Domain-informed Adaptive Sampling for Generalizable PINNs in Metal Additive Manufacturing via Conditional Flow Matching

# Domain-informed Adaptive Sampling for Generalizable PINNs in Metal Additive Manufacturing via Conditional Flow Matching
Source: [https://arxiv.org/html/2610.09126](https://arxiv.org/html/2610.09126)
Hyeonsu LeeAffiliation:Wm Michael Barnes ’64 Department of Industrial and Systems EngineeringAffiliation:Texas A&M UniversityJihoon Jeong††thanks:Corresponding author\.jihoonjeong@tamu\.edu

###### Abstract

Accurate thermal modeling is essential in metal additive manufacturing \(AM\) for understanding the process\-structure\-property chain\. Physics\-informed neural networks \(PINNs\) offer effective surrogate thermal modeling by minimizing physics\-based residual losses at collocation points\. However, prior works typically rely on manually\-crafted, static collocation sampling strategies, which are neither principled nor scalable across process conditions, hindering their generalization capability\. In this work, we provide theoretical analysis through empirical risk minimization, showing that process condition\-aware adaptive sampling is strictly more favorable than conventional static sampling for generalization\. Building on this insight, we propose an adaptive sampling strategy within a two\-stage framework: \(1\) a conditional Flow Matching model that learns approximate high\-residual distributions across different process conditions, and \(2\) a mixed sampling strategy combining this distribution with a domain\-informed base distribution to generate adaptive collocation points for refining the PINN predictor\. Experiments on metal AM numerical benchmarks demonstrate that our method consistently outperforms state\-of\-the\-art PINN baselines, achieving an average 62\.1% reduction in relativeL2L\_\{2\}error under an identical collocation budget, by capturing process\-dependent heat dissipation regions often overlooked in the literature\. To the authors’ knowledge, this is the first adaptive sampling strategy for PINNs in metal AM, contributing to the enhanced generalization and broader applicability\.

## 1Introduction

Metal additive manufacturing \(AM\) has received significant attention for its potential across diverse applications ranging from aerospace, automotive, and agriculture to construction and infrastructure\([Frazier, 2014](https://arxiv.org/html/2610.09126#bib.bib1)\)\. Due to the nature of metal AM processes, its primary advantage lies in its flexibility in realizing complex geometric variations and material compositions tailored to the unique requirements of each application\([Tofail et al\., 2018](https://arxiv.org/html/2610.09126#bib.bib3)\)\. However, process instability and inconsistent part qualification are critical bottlenecks to its widespread industrial adoption\([Svetlizky et al\., 2021](https://arxiv.org/html/2610.09126#bib.bib4)\)\.

To overcome these obstacles, accurate thermal field information is known to be essential in metal AM for understanding defect formation mechanisms and the process\-structure\-property chain\([Smith et al\., 2016](https://arxiv.org/html/2610.09126#bib.bib5);[DebRoy et al\., 2018](https://arxiv.org/html/2610.09126#bib.bib2)\)\. As such, temperature field prediction under unseen process conditions enables effective process optimization, assuring product quality in metal AM\([Ren et al\., 2020](https://arxiv.org/html/2610.09126#bib.bib22)\)\.

Recently, Physics\-Informed Neural Networks \(PINNs\) have been widely adopted as effective thermal surrogate models in metal AM\. PINNs are trained by minimizing physics\-based residuals at sampledcollocation pointsto satisfy governing PDEs, boundary conditions \(BCs\), and initial conditions \(ICs\), thereby yielding physically consistent, interpretable, and reliable predictions\([Raissi et al\., 2019](https://arxiv.org/html/2610.09126#bib.bib19)\)\. Notably, PINNs enable mesh\-free and data\-efficient, or even data\-free training\([Zhu et al\., 2021](https://arxiv.org/html/2610.09126#bib.bib7)\), which has driven extensive research using PINNs for temperature field prediction in metal AM\.

Previous progress in PINNs for metal AM can be broadly categorized into two perspectives: \(1\) improving performance by accounting for steep, highly transient thermal gradients in metal AM processes when training PINNs\([Cooper et al\., 2023](https://arxiv.org/html/2610.09126#bib.bib18);[Peng and Panesar, 2024](https://arxiv.org/html/2610.09126#bib.bib10);[Peng and Panesar, 2025](https://arxiv.org/html/2610.09126#bib.bib11);[Yang et al\., 2026](https://arxiv.org/html/2610.09126#bib.bib16)\), and \(2\) enhancing generalization across unseen process conditions for rapid inference\([Faegh et al\., 2025](https://arxiv.org/html/2610.09126#bib.bib15);[Peng et al\., 2025](https://arxiv.org/html/2610.09126#bib.bib12);[Sajadi et al\., 2025](https://arxiv.org/html/2610.09126#bib.bib13)\)\. Given that metal AM is especially known for its flexibility tailored to diverse applications through varying process conditions including material compositions, prior research has largely focused on advancing the generalizability of PINNs to align with the core flexibility that metal AM promises\.

In this context, recent works have drawn significant attention toparametric PINNsover conventional non\-parametric PINN models, owing to their ability to amortize rapid inference across unseen process conditions\. Unlike non\-parametric PINNs, which often necessitate costly re\-training or transfer learning\-based adaptation for a single, specific target process condition\([Liao et al\., 2023](https://arxiv.org/html/2610.09126#bib.bib9)\), parametric PINNs incorporate PDE parameters as additional inputs, enabling a single trained model to generalize across varying process conditions\. For instance,[Hosseini et al\. \(2023\)](https://arxiv.org/html/2610.09126#bib.bib8)developed a parametric PINN to generalize across different combinations of process parameters \(e\.g\., laser power and scanning speed\) and material properties within a limited range\.[Yuan et al\. \(2025\)](https://arxiv.org/html/2610.09126#bib.bib14)proposed a parametric PINN for generalization to unseen process parameters alongside a transfer learning framework for rapid adaptation to different process conditions\. Most recently,[Lee and Jeong \(2026\)](https://arxiv.org/html/2610.09126#bib.bib20)demonstrated that a decoupled parametric PINN architecture, along with physics\-guided output scaling and a hybrid optimization strategy, achieves strong generalization performance across a wide range of material compositions including both in\-distribution and challenging out\-of\-distribution \(OOD\) regimes\. Their approach outperformed both conventional non\-parametric PINNs and prior parametric PINNs, highlighting the potential of parametric PINNs as an advanced surrogate model for thermal modeling in metal AM\.

These recent advances clearly show that parametric PINNs hold significant potential, not only in their capacity for rapid inference, but also in superior overall generalization performance across unseen process conditions\. Nevertheless, to the best of the authors’ knowledge, one critical limitation remains unaddressed in the literature:

> Althoughcollocation sampling strategiesare critical for both stable convergence and overall predictive performance in PINNs\([Wang et al\., 2022](https://arxiv.org/html/2610.09126#bib.bib17)\), previous research often relies on a single, manually\-crafted collocation sampling strategy that is static and independent of process conditions\([Liao et al\., 2023](https://arxiv.org/html/2610.09126#bib.bib9)\)\. While this strategy is informed by domain knowledge of metal AM and is shown to be adequate to some extent in terms of ensuring training stability and performance\([Faegh et al\., 2025](https://arxiv.org/html/2610.09126#bib.bib15)\), it may be neither effective nor scalable, particularly for parametric PINNs, warranting further investigation\. Specifically, previous approaches statically concentrate collocation points in regions where steep and transient thermal gradients are expected \(e\.g\., near the laser path\), while remaining relatively coarse elsewhere\. However, as evidenced by recent prior work\([Lee and Jeong, 2026](https://arxiv.org/html/2610.09126#bib.bib20)\), high\-residual regions—where predictions deviate significantly from the physics\-based solution—shift substantially depending on process conditions\. For instance, materials with lower thermal conductivity localize intense heat near the laser path, creating sharply concentrated high\-residual zones; conversely, higher thermal conductivity accelerates heat transport, causing high\-residual regions to shift or expand toward broader heat dissipation zones\. In this sense, the conventional static collocation sampling strategy that is independent of process conditions can be suboptimal and ultimately inhibits parametric PINNs from effectively generalizing across diverse process conditions\.

To this end, we introduce a domain\-informed adaptive sampling for parametric PINNs in metal AM\. Specifically, our main contributions are threefold:

1. 1\.Theoretical Analysis on Generalization Error: We provide a theoretical analysis of how collocation sampling impacts generalization across unseen process conditions within an empirical risk minimization framework\. We demonstrate that PINN generalization error decomposes into optimization and statistical errors, where the latter is heavily governed by the variance of the loss estimator\. In this regard, we prove that conventional static sampling strategies are strictly less effective in generalization than adaptive sampling strategies\.
2. 2\.Adaptive Sampling for PINNs in Metal AM using Conditional Flow Matching: Grounded in our theoretical insights, we present an adaptive sampling framework for PINNs in metal AM using Flow Matching\([Lipman et al\., 2022](https://arxiv.org/html/2610.09126#bib.bib21)\)\. By conditioning the generative model on both time steps and PDE parameters, our framework effectively approximates the process condition\-dependent high\-residual distribution and enables efficient generation of collocation points in a single pass\. To the best of the authors’ knowledge, this is the first adaptive sampling framework introduced for PINNs in metal AM\.
3. 3\.Integration of Metal AM Domain Knowledge: We incorporate domain\-informed physical priors both to warm\-start the residual landscape that guides the generative model and to construct a base sampling distribution that is mixed with the learned proposal during training\. This integration stabilizes and accelerates training for both the generator and the predictor\.

We validate the proposed method through extensive experiments on a widely used numerical benchmark, covering five representative metal alloys that span both in\-distribution and OOD regimes\. Under an identical collocation point budget, our method reduces the relativeL2L\_\{2\}error of the current state\-of\-the\-art parametric PINN baseline\([Lee and Jeong, 2026](https://arxiv.org/html/2610.09126#bib.bib20)\)by up to70\.1%70\.1\\%, purely by reallocating where collocation points are sampled\.

Our qualitative analysis further shows that high\-residual regions are not confined to the vicinity of the laser track, as commonly assumed in prior work, but also emerge in heat dissipation regions that evolve with each material’s thermophysical properties\. Capturing these process condition\-dependent shifts, rather than sampling near the laser track alone, proves essential to accurately predicting subsurface and far\-field thermal behavior\. We additionally conduct an ablation and sensitivity analysis, confirming that integrating domain knowledge is critical to achieving both training stability and superior sample efficiency relative to an uninformative uniform base, which reflects the setting where the residual landscape is unknown a priori\. We conclude with a discussion of the limitations of the present study and directions for future work\.

In summary, this work contributes a theoretical foundation for collocation sampling in PINNs from a generalization standpoint, a practical adaptive sampling framework that scales across process conditions via amortized generative modeling, and an integration of domain knowledge that ensures stable and efficient training\. Together, these contributions move parametric PINNs in metal AM closer to strong generalization and rapid inference across diverse process conditions\.

## 2Proposed Method

### 2\.1Nomenclature

We leave the nomenclature in Appendix[A\.1](https://arxiv.org/html/2610.09126#A1.SS1)\.

### 2\.2Problem Setup

LetΩx⊂ℝdx\\Omega\_\{x\}\\subset\\mathbb\{R\}^\{d\_\{x\}\}be the bounded spatial domain representing the part build, and\[0,tend\]⊂ℝ\[0,t\_\{\\mathrm\{end\}\}\]\\subset\\mathbb\{R\}be the time interval over which the AM process is simulated\. The spatiotemporal coordinates thus lie in\(x,t\)∈Ω:=Ωx×\[0,tend\]\(x,t\)\\in\\Omega:=\\Omega\_\{x\}\\times\[0,t\_\{\\mathrm\{end\}\}\], so thatΩ⊂ℝd\\Omega\\subset\\mathbb\{R\}^\{d\}withd=dx\+1d=d\_\{x\}\+1\. Let𝒫⊂ℝm\\mathcal\{P\}\\subset\\mathbb\{R\}^\{m\}be a compact set of PDE parametersμ∈𝒫\\mu\\in\\mathcal\{P\}within known bounds\. In general,μ\\mucan represent any process condition, including, for instance, process parameters \(e\.g\., laser power, scanning speed\) and thermophysical properties of the feedstock material \(e\.g\., density, heat capacity, thermal conductivity\)\. Note that while our theoretical analysis applies to general process conditions, our experiments focus specifically on variations in material properties for benchmarking against previous works\.

Given the governing PDE defined overΩ×𝒫\\Omega\\times\\mathcal\{P\}, parametric PINNs employ a deep neural networkfθ:Ω×𝒫→ℝf\_\{\\theta\}:\\Omega\\times\\mathcal\{P\}\\to\\mathbb\{R\}with learnable parametersθ\\thetato predict the temperature fieldTθ​\(x,t,μ\)∈ℝT\_\{\\theta\}\(x,t;\\mu\)\\in\\mathbb\{R\}\. Training is performed by minimizing a loss function based on the physics\-based residuals\. Here, the PDE residual is given byrθ,pde​\(x,t,μ\)=𝒩⁡\[Tθ\]​\(x,t,μ\)r\_\{\\theta,\\mathrm\{pde\}\}\(x,t;\\mu\)=\\mathcal\{N\}\[T\_\{\\theta\}\]\(x,t;\\mu\), where𝒩⁡\[⋅\]\\mathcal\{N\}\[\\cdot\]is the governing nonlinear differential operator\. Combined with the corresponding BC and IC residuals,rθ,bcr\_\{\\theta,\\mathrm\{bc\}\}andrθ,icr\_\{\\theta,\\mathrm\{ic\}\}, alongside their balancing hyperparametersλpde,λbc\\lambda\_\{\\mathrm\{pde\}\},\\lambda\_\{\\mathrm\{bc\}\}, andλic\\lambda\_\{\\mathrm\{ic\}\}, the total squared residual is formulated as:

rθ​\(x,t,μ\)2:=λpde​rθ,pde2\+λbc​rθ,bc2\+λic​rθ,ic2r\_\{\\theta\}\(x,t;\\mu\)^\{2\}:=\\lambda\_\{\\mathrm\{pde\}\}r\_\{\\theta,\\mathrm\{pde\}\}^\{2\}\+\\lambda\_\{\\mathrm\{bc\}\}r\_\{\\theta,\\mathrm\{bc\}\}^\{2\}\+\\lambda\_\{\\mathrm\{ic\}\}r\_\{\\theta,\\mathrm\{ic\}\}^\{2\}\(1\)In this study, the governing PDE corresponds to the 3D transient heat diffusion equation with Fourier conduction, coupled with convective, radiative, and Gaussian laser\-flux BCs, and an ambient temperature IC \(the detailed formulations are in Appendix[A\.2](https://arxiv.org/html/2610.09126#A1.SS2)\)\.

Our goal is to achieve generalization across the entire spatiotemporal domain and PDE parameter spaceΩ×𝒫\\Omega\\times\\mathcal\{P\}\. We formalize this objective within the standard empirical risk minimization framework by defining two key measures: the population lossℒ⁡\(θ\)\\mathcal\{L\}\(\\theta\)and the empirical lossℒ^N​\(θ,q\)\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)\. First, the population lossℒ⁡\(θ\)\\mathcal\{L\}\(\\theta\)overΩ×𝒫\\Omega\\times\\mathcal\{P\}is defined as:

ℒ⁡\(θ\):=∫Ω×𝒫rθ​\(x,t,μ\)2​p​\(x,t,μ\)​𝑑x​𝑑t​𝑑μ\\mathcal\{L\}\(\\theta\):=\\int\_\{\\Omega\\times\\mathcal\{P\}\}r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\,p\(x,t,\\mu\)\\,dx\\,dt\\,d\\mu\(2\)wherep⁡\(x,t,μ\)p\(x,t,\\mu\)is assumed to be the uniform distribution overΩ×𝒫\\Omega\\times\\mathcal\{P\}\. However, in practice,ℒ⁡\(θ\)\\mathcal\{L\}\(\\theta\)cannot be feasibly computed due to high\-dimensional integration\. Training therefore relies on Monte Carlo estimation with collocation points\{\(xi,ti,μi\)\}i=1N\\\{\(x\_\{i\},t\_\{i\},\\mu\_\{i\}\)\\\}\_\{i=1\}^\{N\}drawn i\.i\.d\. from a proposal densityq⁡\(x,t,μ\)q\(x,t,\\mu\)\. Here, note that we assume both timettand PDE parametersμ\\muare independently drawn from fixed uniform distributions across both training and test evaluations\. Consequently, the joint proposal density factorizes asq⁡\(x,t,μ\)=q⁡\(x∣t,μ\)​p​\(t\)​p​\(μ\)q\(x,t,\\mu\)=q\(x\\mid t,\\mu\)\\,p\(t\)\\,p\(\\mu\)\. Using importance sampling, the empirical loss estimatorℒ^N​\(θ,q\)\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)is then constructed as:

ℒ^N​\(θ,q\):=1N​∑i=1Np⁡\(xi,ti,μi\)q⁡\(xi,ti,μi\)​rθ​\(xi,ti,μi\)2=1N​∑i=1Np⁡\(xi\)q⁡\(xi∣ti,μi\)​rθ​\(xi,ti,μi\)2\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\):=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\frac\{p\(x\_\{i\},t\_\{i\},\\mu\_\{i\}\)\}\{q\(x\_\{i\},t\_\{i\},\\mu\_\{i\}\)\}\\,r\_\{\\theta\}\(x\_\{i\},t\_\{i\};\\mu\_\{i\}\)^\{2\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\frac\{p\(x\_\{i\}\)\}\{q\(x\_\{i\}\\mid t\_\{i\},\\mu\_\{i\}\)\}\\,r\_\{\\theta\}\(x\_\{i\},t\_\{i\};\\mu\_\{i\}\)^\{2\}\(3\)
Throughout this study, we denoteqqas the general spatial proposal\. This includes the conventional static baseline proposalsq⁡\(x∣t\)q\(x\\mid t\), which depend onttto follow the time\-dependent laser track but remain independent ofμ\\mu, and our proposed adaptive proposalqθ​\(x∣t,μ\)q\_\{\\theta\}\(x\\mid t,\\mu\), which dynamically adapts to bothttandμ\\mufor a givenθ\\theta\. Since the target distributions forttandμ\\muare assumed to be fixed uniform across both training and test settings, we aim to design the adaptive spatial conditional distributionqθ​\(x∣t,μ\)q\_\{\\theta\}\(x\\mid t,\\mu\)to minimize the population lossℒ⁡\(θ\)\\mathcal\{L\}\(\\theta\)and thereby achieve effective generalization across unseen process conditions\.

### 2\.3Theoretical analysis of collocation sampling and generalization

In this subsection, we provide a theoretical analysis of how the choice of proposalqqaffects the excess generalization error\. We first show that this error decomposes into an optimization error and a statistical error, with the latter bounded by the variance of the empirical loss estimatorℒ^N​\(θ,q\)\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)\. To lower this bound and thereby minimize the generalization error, we prove that the adaptive proposalqθ\(⋅∣t,μ\)q\_\{\\theta\}\(\\cdot\\mid t,\\mu\)is strictly superior to conventionalμ\\mu\-independent proposalsq\(⋅∣t\)q\(\\cdot\\mid t\)\. Finally, we show that the optimal adaptive proposal is proportional to the squared residual\. Figure[1](https://arxiv.org/html/2610.09126#S2.F1)provides a visual overview of these theoretical results\.

![Refer to caption](https://arxiv.org/html/2610.09126v1/Fig1_Explanation_entire.png)Figure 1:Visual overview of Section[2\.3](https://arxiv.org/html/2610.09126#S2.SS3)\.Left:Proposition[2\.1](https://arxiv.org/html/2610.09126#S2.Thmproposition1)decomposes the generalization error into an optimization error, which is independent of the proposalqq, and a statistical error that scales as𝒪⁡\(σ¯q2/N\)\\mathcal\{O\}\(\\sqrt\{\\bar\{\\sigma\}^\{2\}\_\{q\}/N\}\)\. This implies that reducing the variance of the empirical loss estimator,σ¯q2\\bar\{\\sigma\}^\{2\}\_\{q\}, is a more effective lever for minimizing the statistical error, and thereby the overall generalization error, than merely increasing the total sample sizeNN\.Right:To reduceσ¯q2\\bar\{\\sigma\}^\{2\}\_\{q\}, Theorem[2\.1](https://arxiv.org/html/2610.09126#S2.Thmtheorem1)states that conventionalμ\\mu\-independent static sampling is strictly inferior to adaptive sampling, as its infimum varianceVnc⋆​\(θ\)V^\{\\star\}\_\{\\mathrm\{nc\}\}\(\\theta\)imposes a hard ceiling, whereas theμ\\mu\-conditioned adaptive proposal strictly lowers this variance floor\. Lemma[2\.1](https://arxiv.org/html/2610.09126#S2.Thmlemma1)further characterizes the optimal adaptive proposal as being proportional to the squared residual\.Letθ⋆∈arg⁡minθ∈Θ⁡ℒ⁡\(θ\)\\theta^\{\\star\}\\in\\arg\\min\_\{\\theta\\in\\Theta\}\\mathcal\{L\}\(\\theta\)be the population loss minimizer over the hypothesis classΘ\\Theta,θ^N​\(q\)∈arg⁡minθ∈Θ​ℒ^N​\(θ,q\)\\widehat\{\\theta\}\_\{N\}\(q\)\\in\\arg\\min\_\{\\theta\\in\\Theta\}\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)the empirical loss minimizer under proposalqq, andθ~N\\widetilde\{\\theta\}\_\{N\}the parameter returned after training\. The excess generalization error, the deviation of the population loss of the trained parameterℒ⁡\(θ~N\)\\mathcal\{L\}\(\\widetilde\{\\theta\}\_\{N\}\)from the minimal population lossℒ⁡\(θ⋆\)\\mathcal\{L\}\(\\theta^\{\\star\}\), admits the standard decomposition\([Shalev\-Shwartz and Ben\-David, 2014](https://arxiv.org/html/2610.09126#bib.bib23)\):

ℒ⁡\(θ~N\)−ℒ⁡\(θ⋆\)⏟excess generalization error≤\[ℒ^N​\(θ~N,q\)−ℒ^N​\(θ^N​\(q\),q\)\]⏟optimization error\+2​supθ∈Θ\|ℒ^N​\(θ,q\)−ℒ⁡\(θ\)\|⏟statistical error\\underbrace\{\\mathcal\{L\}\(\\widetilde\{\\theta\}\_\{N\}\)\-\\mathcal\{L\}\(\\theta^\{\\star\}\)\}\_\{\\text\{excess generalization error\}\}\\;\\leq\\;\\underbrace\{\\big\[\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\widetilde\{\\theta\}\_\{N\};q\)\-\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\widehat\{\\theta\}\_\{N\}\(q\);q\)\\big\]\}\_\{\\text\{optimization error\}\}\\;\+\\;2\\,\\sup\_\{\\theta\\in\\Theta\}\\underbrace\{\\big\|\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)\-\\mathcal\{L\}\(\\theta\)\\big\|\}\_\{\\text\{statistical error\}\}\(4\)The detailed derivation of this decomposition is provided in Appendix[A\.3](https://arxiv.org/html/2610.09126#A1.SS3)\. In other words, the excess generalization error is comprised of an optimization error, which is controlled by optimization techniques \(e\.g\., model architecture, optimizer\) and, more importantly, does not depend on the proposalqq, and a statistical error, which depends directly onqqand the sample size of collocation pointsNN\. Thus, reducing this statistical error toward zero is key to achieve true generalization\. The statistical error in turn admits the following variance\-dependent generalization bound:

###### Proposition 2\.1\(Variance\-dependent generalization bound\)\. With probability at least1−δ1\-\\delta, the supremum of the statistical error is bounded as:supθ∈Θ\|ℒ^N​\(θ,q\)−ℒ⁡\(θ\)\|⏟statistical error≤C1​σ¯q2​𝒞​\(Θ,δ\)N\+C2​Mq​R2​𝒞​\(Θ,δ\)N\\sup\_\{\\theta\\in\\Theta\}\\underbrace\{\\big\|\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)\-\\mathcal\{L\}\(\\theta\)\\big\|\}\_\{\\text\{statistical error\}\}\\;\\leq\\;C\_\{1\}\\,\\sqrt\{\\frac\{\\bar\{\\sigma\}\_\{q\}^\{2\}\\,\\mathcal\{C\}\(\\Theta,\\delta\)\}\{N\}\}\\;\+\\;C\_\{2\}\\frac\{M\_\{q\}R^\{2\}\\,\\mathcal\{C\}\(\\Theta,\\delta\)\}\{N\}\(5\)whereσ¯q2:=supθ∈Θσq2​\(θ\)\\bar\{\\sigma\}\_\{q\}^\{2\}:=\\sup\_\{\\theta\\in\\Theta\}\\sigma\_\{q\}^\{2\}\(\\theta\)withσq2​\(θ\):=N​Varq​\[ℒ^N​\(θ,q\)\]\\sigma\_\{q\}^\{2\}\(\\theta\):=N\\,\\mathrm\{Var\}\_\{q\}\\big\[\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)\\big\]denoting the per\-sample variance ofℒ^N​\(θ,q\)\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)\.

The proof is provided in Appendix[A\.4](https://arxiv.org/html/2610.09126#A1.SS4)\. Proposition[2\.1](https://arxiv.org/html/2610.09126#S2.Thmproposition1)delivers the central motivation of this paper: for a conventional static proposalq\(⋅∣t\)q\(\\cdot\\mid t\),σ¯q2\\bar\{\\sigma\}\_\{q\}^\{2\}remains a fixed constant that cannot be changed\. Moreover, merely increasingNNis not computationally efficient as the leading term on the right hand side scales asO⁡\(σ¯q2/N\)O\(\\sqrt\{\\bar\{\\sigma\}\_\{q\}^\{2\}/N\}\)\. For instance, achieving a tenfold reduction in error requires a hundredfold increase inNN\. These observations justify the need for an adaptive sampling proposalqθ\(⋅∣t,μ\)q\_\{\\theta\}\(\\cdot\\mid t,\\mu\), which allowsσq2\\sigma\_\{q\}^\{2\}to be directly minimized, thereby reducing the statistical error\. Therefore, instead of inefficiently increasingNN, an adaptive proposal enables us to effectively minimize the generalization error by directly targeting the constantσ¯q2\\bar\{\\sigma\}\_\{q\}^\{2\}in \([5](https://arxiv.org/html/2610.09126#S2.E5)\), whereas the conventional methods cannot directly reduceσ¯q2\\bar\{\\sigma\}\_\{q\}^\{2\}\.

In this paper, we further provide the theoretical results showing that the conventional static proposal, even the optimal case, is always strictly less effective than the adaptive sampling proposal:

###### Theorem 2\.1\. Let𝒬c=\{q⁡\(x∣t,μ\)\}\\mathcal\{Q\}\_\{\\mathrm\{c\}\}=\\\{q\(x\\mid t,\\mu\)\\\}and𝒬nc=\{q⁡\(x∣t\)\}\\mathcal\{Q\}\_\{\\mathrm\{nc\}\}=\\\{q\(x\\mid t\)\\\}denote the families of conditional and unconditional \(μ\\mu\-independent\) proposal distributions, respectively\. Define the optimal variance bounds asVc⋆​\(θ\):=infq∈𝒬cσq2​\(θ\)V\_\{\\mathrm\{c\}\}^\{\\star\}\(\\theta\):=\\inf\_\{q\\in\\mathcal\{Q\}\_\{\\mathrm\{c\}\}\}\\sigma\_\{q\}^\{2\}\(\\theta\)andVnc⋆​\(θ\):=infq∈𝒬ncσq2​\(θ\)V\_\{\\mathrm\{nc\}\}^\{\\star\}\(\\theta\):=\\inf\_\{q\\in\\mathcal\{Q\}\_\{\\mathrm\{nc\}\}\}\\sigma\_\{q\}^\{2\}\(\\theta\)\. Then, for everyθ∈Θ\\theta\\in\\Theta,Vc⋆​\(θ\)≤Vnc⋆​\(θ\)\.V\_\{\\mathrm\{c\}\}^\{\\star\}\(\\theta\)\\;\\leq\\;V\_\{\\mathrm\{nc\}\}^\{\\star\}\(\\theta\)\.\(6\)

The proofs are provided in Appendix[A\.5](https://arxiv.org/html/2610.09126#A1.SS5)\. In other words, Theorem[2\.1](https://arxiv.org/html/2610.09126#S2.Thmtheorem1)and Remark[2\.1](https://arxiv.org/html/2610.09126#S2.Thmremark1)establish the superiority of conditional sampling over conventional static strategies under theoretical guarantee\. In the conventional static strategies, their independence fromμ\\muimposes a fundamental performance ceilingVnc⋆V\_\{\\mathrm\{nc\}\}^\{\\star\}\. Intuitively, this ceiling is unavoidable because process conditions significantly alter the residual landscapes rather than merely scaling their overall magnitude\. Since no singleμ\\mu\-independent spatial profile can simultaneously capture these parameter\-dependent shape variations, the residual landscape is inherently non\-separable, guaranteeing thatVc⋆​\(θ\)<Vnc⋆​\(θ\)V\_\{\\mathrm\{c\}\}^\{\\star\}\(\\theta\)<V\_\{\\mathrm\{nc\}\}^\{\\star\}\(\\theta\)strictly holds in metal AM thermal modeling\. By conditioning the proposal directly onμ\\mu, our method systematically breaks this static variance barrier and approachesVc⋆V\_\{\\mathrm\{c\}\}^\{\\star\}, enabling robust generalization across diverse process regimes\.

Finally, we now characterize the optimal case of adaptive proposal that attainsVc⋆​\(θ\)V\_\{\\mathrm\{c\}\}^\{\\star\}\(\\theta\)explicitly\. Within the parameter\-conditioned family𝒬c\\mathcal\{Q\}\_\{\\mathrm\{c\}\}, the variance functionalσq2​\(θ\)\\sigma\_\{q\}^\{2\}\(\\theta\)yields a unique minimizer:

###### Lemma 2\.1\(Variance\-optimal adaptive proposal\)\. Fixθ∈Θ\\theta\\in\\Thetawithℒ⁡\(θ\)\>0\\mathcal\{L\}\(\\theta\)\>0\. Within𝒬c\\mathcal\{Q\}\_\{\\mathrm\{c\}\}, the spatial conditional that minimizes the variance functionalσq2​\(θ\)\\sigma\_\{q\}^\{2\}\(\\theta\)is the one that places spatial mass in proportion to the squared residual under each process condition and each instant:qθ⋆​\(x∣t,μ\)∝rθ​\(x,t,μ\)2q^\{\\star\}\_\{\\theta\}\(x\\mid t,\\mu\)\\;\\propto\\;r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\(7\)

The proof is directly from the equality condition of the Cauchy–Schwarz inequality applied in Case 1 of the proof of Theorem[2\.1](https://arxiv.org/html/2610.09126#S2.Thmtheorem1)\(Appendix[A\.5](https://arxiv.org/html/2610.09126#A1.SS5)\)\.

To sum up, minimizing the generalization error is strongly tied to reducing the statistical error, and the effective lever is the variance of the empirical loss estimator rather than merely expanding the sample sizeNN\. In this regard, conventional static,μ\\mu\-independent proposalsq\(⋅∣t\)q\(\\cdot\\mid t\)are strictly worse thanμ\\mu\-conditional proposalsq\(⋅∣t,μ\)q\(\\cdot\\mid t,\\mu\), since the minimal variance of the latter is strictly lower\. Theμ\\mu\-conditional proposal attaining that minimum is proportional to the squared residual under each process condition and instant, which is precisely an adaptive sampling proposal\.

### 2\.4Domain\-Informed Adaptive Sampling via Amortized CFM

In the previous subsection, we demonstrated that theμ\\mu\-conditional adaptive proposalqθ​\(x∣t,μ\)q\_\{\\theta\}\(x\\mid t,\\mu\)is superior to a conventional static proposalq⁡\(x∣t\)q\(x\\mid t\)and that its optimal formqθ⋆​\(x∣t,μ\)q^\{\\star\}\_\{\\theta\}\(x\\mid t,\\mu\)is proportional to the squared residualrθ​\(x,t,μ\)2r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\. Nevertheless, directly implementingqθ⋆q^\{\\star\}\_\{\\theta\}in practice faces the following challenges\. First, calculating the exact normalization constant forqθ⋆q^\{\\star\}\_\{\\theta\}is computationally intractable due to the high\-dimensional integration ofrθ​\(x,t,μ\)2r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\. Second, drawing samples directly from this non\-standard distributionqθ⋆q^\{\\star\}\_\{\\theta\}across everyttand the PDE parameter space𝒫\\mathcal\{P\}is non\-trivial, requiring expensive sampling procedures such as MCMC\. Finally, evaluating the squared residualrθ​\(x,t,μ\)2r\_\{\\theta\}\(x,t;\\mu\)^\{2\}at every sampling iteration introduces significant computational overhead, especially in high\-dimensional domains likeΩ×𝒫⊆ℝd\+m\\Omega\\times\\mathcal\{P\}\\subseteq\\mathbb\{R\}^\{d\+m\}in our case\.

To address these challenges, we propose a domain\-informed adaptive sampling using Flow Matching \(FM\)\. We first use physical priors based on metal AM domain knowledge to warm\-start the initial predictor, obtaining a stable and reliable residual distributionrθsrcr\_\{\\theta\_\{\\mathrm\{src\}\}\}\. We then use Flow Matching to approximate the target residual distribution as a continuous function and condition FM on both timettand process conditionsμ\\mu, which we refer to as Conditional Flow Matching \(CFM\), to alleviate the curse of dimensionality\. Finally, we blend the domain\-informed base with the learned high\-residual proposal to handle the exploration–exploitation trade\-off, and finally trainfθf\_\{\\theta\}with the refined collocation points\.

This approach reduces computational overhead and mitigates the curse of dimensionality\. Flow Matching enables continuous density modeling, serving as a scalable alternative to traditional sampling algorithms\. By further training the FM model conditioned on both timettand process conditionsμ\\mu, the sampling process is amortized, allowing direct sample generation from high\-residual distributions for each process conditionμ\\muin a single forward pass\. This streamlines the training pipeline and provides representative collocation points per process condition, enabling the predictorfθf\_\{\\theta\}to minimize generalization error across unseen process conditions–our primary objective\.

#### 2\.4\.1Flow matching background

We briefly review Flow Matching \(FM\), the deep generative backbone used in this study\. Flow Matching\([Lipman et al\., 2022](https://arxiv.org/html/2610.09126#bib.bib21)\)is a likelihood\-free training criterion for continuous normalizing flows designed to learn a time\-dependent vector field that transports samples from a tractable source distribution to a desired target distribution\. Specifically, let\{pτ\}τ∈\[0,1\]\\\{p\_\{\\tau\}\\\}\_\{\\tau\\in\[0,1\]\}be a family of probability densities onℝd\\mathbb\{R\}^\{d\}interpolating from a source densityp0p\_\{0\}atτ=0\\tau=0to a target densityp1p\_\{1\}atτ=1\\tau=1\. A time\-dependent vector fieldv:\[0,1\]×ℝd→ℝdv:\[0,1\]\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}generates this probability path if integrating the velocity fieldz˙τ=v⁡\(τ,zτ\)\\dot\{z\}\_\{\\tau\}=v\(\\tau,z\_\{\\tau\}\)starting fromz0∼p0z\_\{0\}\\sim p\_\{0\}yieldszτ∼pτz\_\{\\tau\}\\sim p\_\{\\tau\}for everyτ∈\[0,1\]\\tau\\in\[0,1\]\. To construct this vector field, Flow Matching regresses a parameterized vector fieldvϕv\_\{\\phi\}onto the target displacement vectors between paired source and target samples\. Given a coupling\(x\(0\),x\(1\)\)∼γ\(x^\{\(0\)\},x^\{\(1\)\}\)\\sim\\gammawith marginalsp0p\_\{0\}andp1p\_\{1\}, the linear interpolantx\(τ\)=\(1−τ\)​x\(0\)\+τ​x\(1\)x^\{\(\\tau\)\}=\(1\-\\tau\)x^\{\(0\)\}\+\\tau x^\{\(1\)\}thus yields a constant velocity fieldx˙\(τ\)=x\(1\)−x\(0\)\\dot\{x\}^\{\(\\tau\)\}=x^\{\(1\)\}\-x^\{\(0\)\}\.

A key advantage of Flow Matching lies in its straightforward optimization trajectory\. Training Flow Matching involves a simple mean\-squared error \(MSE\) regression while other generative models are prone to exhibit optimization challenges \(e\.g\., GANs\)\([Arjovsky and Bottou, 2017](https://arxiv.org/html/2610.09126#bib.bib24)\)\. Therefore, it enables rapid conditional sample generation while maintaining training stability across high\-dimensional parameter spaces\.

#### 2\.4\.2Training procedure

![Refer to caption](https://arxiv.org/html/2610.09126v1/Fig2_method_cfm_transport_mixture_3d.png)Figure 2:Domain\-informed adaptive collocation via CFM\.Top\(a–d\): CFM transport of uniform base samples \(τ=0\\tau=0\) into the adaptive proposalqθsrc⋆\(⋅∣t,μ\)q^\{\\star\}\_\{\\theta\_\{\\mathrm\{src\}\}\}\(\\cdot\\mid t,\\mu\)\(τ=1\\tau=1\), which concentrates in high\-residual regions\.Bottom: \(e\) Domain\-informed baseb\(⋅∣t\)b\(\\cdot\\mid t\); \(f–h\) Training mixtureqαbq\_\{\\alpha\}^\{b\}att=0\.5,1\.5t=0\.5,1\.5, and2\.5​s2\.5\\,\\text\{s\}, where the base remains fixed while the CFM component tracks high\-residual regions \(visualized for Ti\-6Al\-4V\)\.Our proposed domain\-informed adaptive sampling framework adopts a two\-stage training scheme\. Stage 1 trains a CFM generatorgϕg\_\{\\phi\}to amortize the variance\-optimal proposalqθsrc⋆​\(x∣t,μ\)q^\{\\star\}\_\{\\theta\_\{\\mathrm\{src\}\}\}\(x\\mid t,\\mu\), while Stage 2 trains the final predictorfθf\_\{\\theta\}using a domain\-informed mixture proposal that balances exploration and exploitation\. The overall training procedure is summarized in Algorithm[1](https://arxiv.org/html/2610.09126#alg1)\.

Algorithm 1Domain\-Informed Adaptive Sampling via Conditional Flow Matching1:domain\-informed base

b\(⋅∣t\)b\(\\cdot\\mid t\); mixture weight

α\\alpha; number of stored conditions

JJ; pool size

GG; resample size

MresM\_\{\\mathrm\{res\}\}; iteration counts

KwarmK\_\{\\mathrm\{warm\}\},

KflowK\_\{\\mathrm\{flow\}\},

KPINNK\_\{\\mathrm\{PINN\}\}; batch sizes

BB\(CFM\) and

NN\(PINN\); ODE steps

NODEN\_\{\\mathrm\{ODE\}\}\.

2:

3:Stage 1: train the amortized generatorgϕg\_\{\\phi\}

4:\(Warm\-start\)Train the source predictor

fθsrcf\_\{\\theta\_\{\\mathrm\{src\}\}\}for

KwarmK\_\{\\mathrm\{warm\}\}steps on the residual loss \([1](https://arxiv.org/html/2610.09126#S2.E1)\), drawing

tt,

μ\\muuniformly and

x∼b\(⋅∣t\)x\\sim b\(\\cdot\\mid t\); then freeze

θsrc\\theta\_\{\\mathrm\{src\}\}\.

5:\(Score\)For each of

JJuniformly drawn conditions

μj\\mu\_\{j\}: draw a pool

𝒟j=\{\(x~k,t~k\)\}k=1G\\mathcal\{D\}\_\{j\}=\\\{\(\\tilde\{x\}\_\{k\},\\tilde\{t\}\_\{k\}\)\\\}\_\{k=1\}^\{G\}uniformly over

Ω\\Omegaand score every element by

sj,k←rθsrc​\(x~k,t~k,μj\)2s\_\{j,k\}\\leftarrow r\_\{\\theta\_\{\\mathrm\{src\}\}\}\(\\tilde\{x\}\_\{k\},\\tilde\{t\}\_\{k\};\\mu\_\{j\}\)^\{2\}\.

6:for

KflowK\_\{\\mathrm\{flow\}\}stepsdo⊳\\trianglerightCFM training

7:Pick

BBconditions among the

JJstored ones; for each, resample one flow target

\(x\(1\),t†\)\(x^\{\(1\)\},t^\{\\dagger\}\)from

𝒟j\\mathcal\{D\}\_\{j\}via \([8](https://arxiv.org/html/2610.09126#S2.E8)\) and pair it with a source draw

x\(0\)∼p0x^\{\(0\)\}\\sim p\_\{0\}\.

8:Update

ϕ\\phiby one gradient step on

ℒCFM​\(ϕ\)\\mathcal\{L\}\_\{\\mathrm\{CFM\}\}\(\\phi\)in \([9](https://arxiv.org/html/2610.09126#S2.E9)\)\.

9:endfor

10:

11:Stage 2: train the final predictorfθf\_\{\\theta\}with the frozen generator

12:Initialize

fθf\_\{\\theta\}from scratch\.

13:for

KPINNK\_\{\\mathrm\{PINN\}\}stepsdo

14:Draw

tit\_\{i\},

μi\\mu\_\{i\}uniformly and

xi∼qαb\(⋅∣ti,μi\)x\_\{i\}\\sim q\_\{\\alpha\}^\{b\}\(\\cdot\\mid t\_\{i\},\\mu\_\{i\}\)from the mixture \([10](https://arxiv.org/html/2610.09126#S2.E10)\) for

i=1,…,Ni=1,\\dots,N; samples from the

gϕg\_\{\\phi\}component are obtained by integrating

vϕv\_\{\\phi\}with

NODEN\_\{\\mathrm\{ODE\}\}forward\-Euler steps from

x\(0\)∼p0x^\{\(0\)\}\\sim p\_\{0\}\.

15:Update

θ\\thetaby one gradient step on the residual loss

1N​∑i=1Nrθ​\(xi,ti,μi\)2\\tfrac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}r\_\{\\theta\}\(x\_\{i\},t\_\{i\};\\mu\_\{i\}\)^\{2\}\.

16:endfor

17:returnthe generator

gϕg\_\{\\phi\}and the predictor

fθf\_\{\\theta\}\.

##### Stage 1: Warm\-start & Amortized Generator Training

Letb⁡\(x∣t\)b\(x\\mid t\)denote aμ\\mu\-independent, domain\-informed base density \(visualized in Figure[2](https://arxiv.org/html/2610.09126#S2.F2)\(e\)\)\. Collocation points are initially drawn frombbto warm\-start the predictorfθsrcf\_\{\\theta\_\{\\mathrm\{src\}\}\}, until it yields physics\-consistent solutions with a reliable residual distributionrθsrcr\_\{\\theta\_\{\\mathrm\{src\}\}\}\.

We now apply CFM to our adaptive sampling setting\. Here, our goal is to learn a velocity fieldvϕv\_\{\\phi\}transporting a simple source densityp0p\_\{0\}onΩx\\Omega\_\{x\}to the variance\-optimal targetqθsrc⋆\(⋅∣t,μ\)q^\{\\star\}\_\{\\theta\_\{\\mathrm\{src\}\}\}\(\\cdot\\mid t,\\mu\)\. We letp0p\_\{0\}be uniform onΩx\\Omega\_\{x\}, which is convenient to sample while able to reach any high\-residual region\. Note that the flow\-matching timeτ∈\[0,1\]\\tau\\in\[0,1\]is entirely distinct from the physical timett, which enters only as a conditioning input held fixed along the path\. In practice, collocation points for the PDE, BC, and IC residuals are drawn on different subdomains; we therefore additionally conditionvϕv\_\{\\phi\}on a residual\-group identifierℓ∈\{pde,bc,ic\}\\ell\\in\\\{\\mathrm\{pde\},\\mathrm\{bc\},\\mathrm\{ic\}\\\}, which we omit from the notation below for brevity\.

To construct the targets, we freeze the source parametersθsrc\\theta\_\{\\mathrm\{src\}\}and work one condition at a time\. For each ofJJuniformly sampled process conditions fromμj∼𝒫\\mu\_\{j\}\\sim\\mathcal\{P\}, we draw a pool𝒟j=\{\(x~j,k,t~j,k\)\}k=1G\\mathcal\{D\}\_\{j\}=\\\{\(\\tilde\{x\}\_\{j,k\},\\tilde\{t\}\_\{j,k\}\)\\\}\_\{k=1\}^\{G\}uniformly overΩ\\Omega, score it bysj,k=rθsrc​\(x~j,k,t~j,k,μj\)2s\_\{j,k\}=r\_\{\\theta\_\{\\mathrm\{src\}\}\}\(\\tilde\{x\}\_\{j,k\},\\tilde\{t\}\_\{j,k\};\\mu\_\{j\}\)^\{2\}, and resampleMresM\_\{\\mathrm\{res\}\}elements:

\{\(xj,i†,tj,i†\)\}i=1Mres​∼i\.i\.d\.​Multinomial​\(𝒟j,wj,k∝sj,k\),j=1,…,J\.\\\{\(x\_\{j,i\}^\{\\dagger\},\\,t\_\{j,i\}^\{\\dagger\}\)\\\}\_\{i=1\}^\{M\_\{\\mathrm\{res\}\}\}\\overset\{\\text\{i\.i\.d\.\}\}\{\\sim\}\\mathrm\{Multinomial\}\\\!\\big\(\\mathcal\{D\}\_\{j\};\\;w\_\{j,k\}\\propto s\_\{j,k\}\\big\),\\qquad j=1,\\dots,J\.\(8\)Uniform candidates make the weights proportional to the residual itself, with no proposal density to estimate\. We then split each draw, conditioning on\(μj,tj,i†\)\(\\mu\_\{j\},t\_\{j,i\}^\{\\dagger\}\)and takingxj,i†x\_\{j,i\}^\{\\dagger\}as the flow target; because the weights are normalized within a condition, this recovers exactlyqθsrc⋆\(⋅∣t,μ\)q^\{\\star\}\_\{\\theta\_\{\\mathrm\{src\}\}\}\(\\cdot\\mid t,\\mu\)of Lemma[2\.1](https://arxiv.org/html/2610.09126#S2.Thmlemma1)\. Per\-condition normalization also fixes everyμj\\mu\_\{j\}atMresM\_\{\\mathrm\{res\}\}targets, so the parameter marginal is uniform by construction\.

Pairing each target with an independent source drawxi\(0\)∼p0x\_\{i\}^\{\(0\)\}\\sim p\_\{0\}, we trainvϕv\_\{\\phi\}via stochastic gradient descent on

ℒCFM​\(ϕ\)=𝔼τ∼U⁡\[0,1\]​𝔼\(t,μ,x\(0\),x\(1\)\)​\[‖vϕ​\(τ,x\(τ\),t,μ\)−\(x\(1\)−x\(0\)\)‖2\],\\mathcal\{L\}\_\{\\mathrm\{CFM\}\}\(\\phi\)=\\mathbb\{E\}\_\{\\tau\\sim U\[0,1\]\}\\,\\mathbb\{E\}\_\{\(t,\\,\\mu,\\,x^\{\(0\)\},\\,x^\{\(1\)\}\)\}\\\!\\Big\[\\big\\\|v\_\{\\phi\}\\\!\\big\(\\tau,\\,x^\{\(\\tau\)\},\\,t,\\,\\mu\\big\)\-\\big\(x^\{\(1\)\}\-x^\{\(0\)\}\\big\)\\big\\\|^\{2\}\\Big\],\(9\)wherex\(τ\):=\(1−τ\)​x\(0\)\+τ​x\(1\)x^\{\(\\tau\)\}:=\(1\-\\tau\)x^\{\(0\)\}\+\\tau\\,x^\{\(1\)\}is the linear interpolant andx\(1\)=xi†x^\{\(1\)\}=x\_\{i\}^\{\\dagger\}\. The trainedvϕv\_\{\\phi\}then defines the generatorgϕ\(⋅∣t,μ\)≈qθsrc⋆\(⋅∣t,μ\)g\_\{\\phi\}\(\\cdot\\mid t,\\mu\)\\approx q^\{\\star\}\_\{\\theta\_\{\\mathrm\{src\}\}\}\(\\cdot\\mid t,\\mu\)as the law ofx\(1\)x^\{\(1\)\}obtained by integratingdd​τ​x\(τ\)=vϕ​\(τ,x\(τ\),t,μ\)\\tfrac\{d\}\{d\\tau\}x^\{\(\\tau\)\}=v\_\{\\phi\}\(\\tau,x^\{\(\\tau\)\},t,\\mu\)fromx\(0\)∼p0x^\{\(0\)\}\\sim p\_\{0\}toτ=1\\tau=1, which we approximate with the forward Euler scheme usingNODE=10N\_\{\\mathrm\{ODE\}\}=10\.

##### Stage 2: Predictor Training via Domain\-Informed Mixture

In Stage 2, we freeze the trained CFM generatorgϕg\_\{\\phi\}and initialize the final predictorfθf\_\{\\theta\}from scratch\. In contrast to Stage 1, the collocation pointsxxis drawn from both the learned proposalgϕg\_\{\\phi\}and the basebbwith the proposed domain\-informed mixture proposalqαbq\_\{\\alpha\}^\{b\}:

qαb​\(x∣t,μ\):=\(1−α\)​gϕ​\(x∣t,μ\)\+α​b​\(x∣t\)q\_\{\\alpha\}^\{b\}\(x\\mid t,\\mu\):=\(1\-\\alpha\)\\,g\_\{\\phi\}\(x\\mid t,\\mu\)\+\\alpha\\,b\(x\\mid t\)\(10\)where the mixture weightα\\alphacontrols the exploration–exploitation trade\-off: smallerα\\alphamore aggressively exploits the learned high\-residual regions, while largerα\\alpharetains broader coverage that defends against estimation error inrθr\_\{\\theta\}\. The predictor parameterθ\\thetaare then updated on the residual loss evaluated at collocation points drawn fromqαbq\_\{\\alpha\}^\{b\}\. This prevents the risk that exclusive sampling from the learned high\-residual proposalgϕg\_\{\\phi\}neglects underexplored regions wherefθf\_\{\\theta\}initially underestimates the residual\.

## 3Experiments

##### System Description

To evaluate the proposed method, we adopt a widely recognized numerical benchmark\([Liao et al\., 2023](https://arxiv.org/html/2610.09126#bib.bib9);[Faegh et al\., 2025](https://arxiv.org/html/2610.09126#bib.bib15);[Lee and Jeong, 2026](https://arxiv.org/html/2610.09126#bib.bib20)\)based on a bare\-plate laser scanning example representative of metal AM processes\. In this benchmark, a moving laser heat source scans the surface of a metal substrate, generating highly localized, transient, and rapid thermal gradients governed by laser heating, conductive heat transfer within the substrate, and convective/radiative heat losses at the boundaries\. Representative snapshots of the resulting temperature field are illustrated in Figure[3](https://arxiv.org/html/2610.09126#S3.F3)\. To ensure a fair comparison, all simulation setups and process parameters follow the previous research\([Lee and Jeong, 2026](https://arxiv.org/html/2610.09126#bib.bib20)\)\.

![Refer to caption](https://arxiv.org/html/2610.09126v1/Fig3_ti64_snapshots.png)Figure 3:Ground\-truth temperature field snapshots att=0\.5t=0\.5,1\.51\.5, and2\.5​s2\.5\\,\\mathrm\{s\}obtained via high\-fidelityJAX\-AMsimulation for the bare\-plate scanning benchmark \(visualized using Ti–6Al–4V\)\. The thermal evolution captures the localized moving laser heat source and subsequent heat diffusion\.
##### Process conditionsμ\\mu\.

Following the experimental design in[Lee and Jeong \(2026\)](https://arxiv.org/html/2610.09126#bib.bib20), we consider five materials widely used in metal AM as unseen process conditionsμ\\muto assess the generalizability of the proposed framework\. It is noteworthy that variations in each material’s thermophysical properties substantially alter its thermal behavior, inducing significant shifts in the underlying residual landscapes\. We define the continuous process condition space𝒫\\mathcal\{P\}over the characteristic ranges of densityρc\\rho\_\{c\}, heat capacityCpC\_\{p\}, and thermal conductivitykck\_\{c\}:

𝒫:=\[ρc,min,ρc,max\]×\[Cp,min,Cp,max\]×\[kc,min,kc,max\]\\mathcal\{P\}:=\[\\rho\_\{c,\\text\{min\}\},\\rho\_\{c,\\text\{max\}\}\]\\times\[C\_\{p,\\text\{min\}\},C\_\{p,\\text\{max\}\}\]\\times\[k\_\{c,\\text\{min\}\},k\_\{c,\\text\{max\}\}\]where the boundaries are set toρc∈\[3000,10000\]​kg/m3\\rho\_\{c\}\\in\[3000,10000\]\\,\\mathrm\{kg/m\}^\{3\},Cp∈\[300,1000\]​J/\(kg⋅K\)C\_\{p\}\\in\[300,1000\]\\,\\mathrm\{J/\(kg\\cdot K\)\}, andkc∈\[3,100\]​W/\(m⋅K\)k\_\{c\}\\in\[3,100\]\\,\\mathrm\{W/\(m\\cdot K\)\}, identically applied at both training and testing phases as formulated in Section[2\.2](https://arxiv.org/html/2610.09126#S2.SS2)\. We evaluate five representative materials with properties denoted as\(ρc,Cp,kc\)\(\\rho\_\{c\},C\_\{p\},k\_\{c\}\):

- •Ti–6Al–4V\(4430,560,6\.7\)\(4430,560,6\.7\): Standard aerospace alloy characterized by a high strength\-to\-weight ratio\.
- •Inconel 718\(8220,435,11\.4\)\(8220,435,11\.4\): Nickel\-based superalloy featuring high\-temperature thermal stability\.
- •SS 316L\(8000,500,16\.0\)\(8000,500,16\.0\): Austenitic stainless steel with high corrosion resistance\.
- •AlSi10Mg\(2670,950,150\)\(2670,950,150\): Lightweight structural alloy whose thermal conductivitykck\_\{c\}exceeds the upper limit of𝒫\\mathcal\{P\}by1\.5×1\.5\\times\.
- •Copper\(8960,385,401\)\(8960,385,401\): Highly conductive material with an extreme thermal conductivitykck\_\{c\}exceeding the upper bound of𝒫\\mathcal\{P\}by more than4×4\\times\.

Here, the first three materials represent in\-distribution \(ID\) cases, whereas AlSi10Mg and Copper serve as OOD extrapolation cases as their thermal conductivitieskck\_\{c\}substantially exceed the training upper boundkc,maxk\_\{c,\\text\{max\}\}\. Note that OOD evaluation is known to be notoriously challenging compared to ID cases, allowing us to validate whether the extrapolation capability demonstrated in prior work\([Lee and Jeong, 2026](https://arxiv.org/html/2610.09126#bib.bib20)\)remains effective under our adaptive sampling scheme\.

##### Baselines and Evaluation Metric

We evaluate the proposed method against two PINN baselines\.

- •Non\-parametric PINN \(N\-PINN\)\([Liao et al\., 2023](https://arxiv.org/html/2610.09126#bib.bib9)\): Conventional PINN formulation where material properties are treated as fixed constants\. A separate network must be trained from scratch for each material condition\.
- •Advanced Parametric PINN \(AP\-PINN\)\([Lee and Jeong, 2026](https://arxiv.org/html/2610.09126#bib.bib20)\): State\-of\-the\-art parametric PINN for material\-agnostic thermal modeling in metal AM, trained using a static domain\-informed collocation strategyb⁡\(x∣t\)b\(x\\mid t\)\. Our predictorfθf\_\{\\theta\}adopts this architecture as our backbone without modification, ensuring that any performance gain is entirely attributable to the proposed adaptive sampling strategy\.

For quantitative comparison, we report the relativeL2L\_\{2\}error, a standard metric in the PINN literature\([Raissi et al\., 2019](https://arxiv.org/html/2610.09126#bib.bib19)\)\. This metric measures the normalized discrepancy between the predicted temperature fieldTθT\_\{\\theta\}and the ground\-truth solutionTTover the uniformly discretized spatiotemporal domainΩ\\Omega\. For a given process conditionμ\\mu, the relativeL2L\_\{2\}error is defined as:

RelativeL2Error\(%\)=∑n=1NFEM\|T⁡\(x\(n\),t\(n\),μ\)−Tθ​\(x\(n\),t\(n\),μ\)\|2∑n=1NFEM\|T⁡\(x\(n\),t\(n\),μ\)\|2×100\\text\{Relative \}L\_\{2\}\\text\{ Error\}\(\\%\)=\\frac\{\\sqrt\{\\sum\_\{n=1\}^\{N\_\{\\text\{FEM\}\}\}\\left\|T\\left\(x^\{\(n\)\},t^\{\(n\)\};\\mu\\right\)\-T\_\{\\theta\}\\left\(x^\{\(n\)\},t^\{\(n\)\};\\mu\\right\)\\right\|^\{2\}\}\}\{\\sqrt\{\\sum\_\{n=1\}^\{N\_\{\\text\{FEM\}\}\}\\left\|T\\left\(x^\{\(n\)\},t^\{\(n\)\};\\mu\\right\)\\right\|^\{2\}\}\}\\times 100Here,TTrepresents the reference ground\-truth temperature field obtained viaJAX\-AM111[https://github\.com/CMSL\-HKUST/jax\-am](https://github.com/CMSL-HKUST/jax-am)\([Xue et al\., 2023](https://arxiv.org/html/2610.09126#bib.bib6)\), an open\-source library for high\-fidelity FEM simulations in metal AM\.NFEMN\_\{\\text\{FEM\}\}denotes the total number of nodes in the FEM simulations, which isNFEM=88,939N\_\{\\text\{FEM\}\}=88\{,\}939in this study\.

##### Implementation Details

To maintain consistency across all models, baselines are implemented following their original hyperparameters and training configurations\. The loss weights are set to\(λpde,λbc,λic\)=\(1,1,10−4\)\(\\lambda\_\{\\mathrm\{pde\}\},\\lambda\_\{\\mathrm\{bc\}\},\\lambda\_\{\\mathrm\{ic\}\}\)=\(1,1,10^\{\-4\}\)following[Liao et al\. \(2023\)](https://arxiv.org/html/2610.09126#bib.bib9);[Lee and Jeong \(2026\)](https://arxiv.org/html/2610.09126#bib.bib20)\. Both baselines adopt the same domain\-informed static collocation sampling strategy, and we adopt their defined base distributionb⁡\(x∣t\)b\(x\\mid t\)for our method as well\. Furthermore, the total collocation point budgetNNand its per\-group allocations \(Npde,Nbc,NicN\_\{\\text\{pde\}\},N\_\{\\text\{bc\}\},N\_\{\\text\{ic\}\}\) strictly match the baseline setup, guaranteeing an identical sampling budget across all methods\.

For our sampling scheme, the CFM velocity networkvϕv\_\{\\phi\}is parameterized using33hidden layers of width128128with SiLU activations, encoding the flow\-matching timeτ\\tauvia a3232\-dimensional sinusoidal embedding\. The generatorgϕg\_\{\\phi\}is a substantially smaller network relative to our predictorfθf\_\{\\theta\}, which is identical to the predictor of the AP\-PINN baseline\. The mixture weight is fixed toα=0\.6\\alpha=0\.6, and its hyperparameter sensitivity is discussed in the following subsection\. Note that training the CFM generator takes approximately22minutes on a single GPU and drawing adaptive samples adds less than1%1\\%to the per\-run training time relative to trainingfθf\_\{\\theta\}\.

All quantitative results are reported as the mean±\\pmstandard deviation computed across five random seeds\. Experiments were conducted on a single NVIDIA RTX 5090 GPU using Python 3\.10 and PyTorch 2\.9\. The complete implementation, including data and code are available at[https://github\.com/hsleecri/AdaptiveSamplingPINN\-AM](https://github.com/hsleecri/AdaptiveSamplingPINN-AM)\.

## 4Results and Discussion

##### Quantitative Evaluation

Table[1](https://arxiv.org/html/2610.09126#S4.T1)summarizes the quantitative comparison between the baselines and our proposed method\. \(Qualitative temperature fields from our predictions and FEM ground\-truth solutions att=0\.5​s,1\.5​st=0\.5\\text\{ s\},1\.5\\text\{ s\}, and2\.5​s2\.5\\text\{ s\}are provided in Appendix[A\.7](https://arxiv.org/html/2610.09126#A1.SS7)\.\) While the state\-of\-the\-art parametric PINN baseline \(AP\-PINN\) outperforms the non\-parametric baseline \(N\-PINN\) across five material spanning both ID and OOD regimes, our proposed method considerably outperforms both baselines\. Specifically, our method further reduces the relativeL2L\_\{2\}error of AP\-PINN by an average of62\.1%62\.1\\%and up to70\.1%70\.1\\%, achieving consistent gains across all materials\.

Table 1:RelativeL2L\_\{2\}error \[%\] \(mean±\\pmstd over five seeds\) across five materials\.*Average*denotes the mean across all five materials\. The last row reports the error reduction of the proposed method relative to the AP\-PINN baseline\. Lower is better\.![Refer to caption](https://arxiv.org/html/2610.09126v1/Fig4_swapped_training_dynamics.png)Figure 4:Training dynamics \(mean±\\pmstd over five seeds\): \(a\) average relativeL2L\_\{2\}error on the five materials, and \(b–d\) the PDE, BC, and IC training losses evaluated on each method’s own sampled collocation points\. The proposed method reaches a lower relativeL2L\_\{2\}error despite higher training losses on the three losses, which is a direct consequence of concentrating collocation points in high\-residual regions\.
##### Training Loss Analysis

To discover where this superiority stems from, Figure[4](https://arxiv.org/html/2610.09126#S4.F4)illustrates the training dynamics of AP\-PINN alongside our proposed method\. Interestingly, the PDE, BC, and IC training losses of the proposed method are consistently higher than those of AP\-PINN, whereas our relativeL2L\_\{2\}error is substantially lower\. We attribute this phenomenon to the expected signature of adaptive collocation: our adaptive sampling proposal concentrates a higher proportion of points in high\-residual regions, naturally leading to a larger residual over the sampled points\.

Specifically, after the transition of optimizers from Adam to stochastic L\-BFGS \(gray dashed line\), our model starts to minimize the three losses simultaneously, indicating its convergence toward a good local optimum\. During this stage, PDE and BC losses are dramatically reduced, and the relativeL2L\_\{2\}error rapidly decreases at the same time\. This indicates that, despite these higher evaluated residuals, the strategically placed collocation points allow the model to learn the underlying thermodynamics far more effectively across both the spatiotemporal domain and PDE parameter space\. Consequently, effective training on points sampled from high\-residual regions ultimately yields significantly lower relativeL2L\_\{2\}errors\.

![Refer to caption](https://arxiv.org/html/2610.09126v1/Fig5_swapped_loss_empirical_vs_population.png)Figure 5:Comparison of empirical lossℒ^N​\(θ,q\)\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)\([3](https://arxiv.org/html/2610.09126#S2.E3)\) and population lossℒ⁡\(θ\)\\mathcal\{L\}\(\\theta\)\([2](https://arxiv.org/html/2610.09126#S2.E2)\) \(mean±\\pmstd over five seeds\): \(a\) training evolution and \(b\) final values\. The proposed method exhibits a higher empirical loss during training, yet consistently achieves a lower population loss by focusing samples on high\-residual regions, which leads to more effective learning\.To substantiate this claim, we re\-evaluate the training dynamics using the weighted sum of residuals \([1](https://arxiv.org/html/2610.09126#S2.E1)\) computed on collocation points uniformly sampled overΩ×𝒫\\Omega\\times\\mathcal\{P\}, which serves as an unbiased estimator of the true population lossℒ⁡\(θ\)\\mathcal\{L\}\(\\theta\), rather than relying on the empirical loss estimatorℒ^N​\(θ,q\)\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)evaluated under each method’s respective proposal distribution \(i\.e\., the learned proposalqαb​\(x∣t,μ\)q^\{b\}\_\{\\alpha\}\(x\\mid t,\\mu\)for our method and the static domain\-informed proposalb⁡\(x∣t\)b\(x\\mid t\)for AP\-PINN\)\. As shown in Figure[5](https://arxiv.org/html/2610.09126#S4.F5), the proposed method exhibits a substantially lower estimated population loss compared to AP\-PINN\. Conversely, the empirical loss evaluated on its own proposal is higher for our method\. This confirms that the proposed adaptive proposal concentrates collocation points in high\-residual regions during training, ultimately leading to superior generalization performance\.

![Refer to caption](https://arxiv.org/html/2610.09126v1/Fig6a_swapped_thermal_history_temperature.png)\(a\)Thermal histories predicted by AP\-PINN and ours
![Refer to caption](https://arxiv.org/html/2610.09126v1/Fig6b_swapped_thermal_history_deviation.png)\(b\)Deviations from FEM by AP\-PINN and ours

Figure 6:Comparison of thermal histories at increasing depths below the laser scan track across five materials \(columns\) at the top surface,2​mm2\\,\\mathrm\{mm\}, and4​mm4\\,\\mathrm\{mm\}depth \(rows\) \(mean±\\pmstd over five seeds\): \(a\) predicted temperature profiles and FEM reference and \(b\) temperature deviationsΔ​T=Tθ−T\\Delta T=T\_\{\\theta\}\-Trelative to the FEM reference\. Both methods match well at the top surface under direct laser scanning, but the AP\-PINN baseline underperforms significantly in late\-time and far\-field regions where heat dissipation governs over laser irradiation\. Ultimately, AP\-PINN misses the subsurface thermal profiles entirely at4​mm4\\,\\mathrm\{mm\}depth across all materials, whereas the proposed method accurately captures it\.
##### Thermal History Comparison

To examine how these reductions in both residuals and relativeL2L\_\{2\}error translate to temperature field predictions, we compare the thermal histories predicted by AP\-PINN and the proposed method against the ground\-truth FEM solutions in Figure[6a](https://arxiv.org/html/2610.09126#S4.F6.sf1), alongside their corresponding temperature deviations in Figure[6b](https://arxiv.org/html/2610.09126#S4.F6.sf2)\. This comparison is evaluated at three specific positions below the scan track \(x=10​mmx=10\\,\\mathrm\{mm\}\) to capture the distinct thermal behavior of each material: at the top surface \(z=0​mmz=0\\,\\mathrm\{mm\}\),2​mm2\\,\\mathrm\{mm\}below, and4​mm4\\,\\mathrm\{mm\}below the top surface \(which corresponds to2​mm2\\text\{ mm\}upward from the bottom surface\)\.

As shown in Figure[6a](https://arxiv.org/html/2610.09126#S4.F6.sf1), both methods closely match the FEM reference at the top surface\. They both successfully capture the rapid transient thermal gradients induced by laser irradiation across all materials, as well as accounting for their distinct temperature magnitudes driven by varying thermophysical properties\. At2​mm2\\,\\mathrm\{mm\}depth, however, deviations begin to emerge as differences in thermophysical properties lead to distinct thermal conduction patterns\. With increasing depth, the predictions diverge significantly\. At4​mm4\\,\\mathrm\{mm\}depth, AP\-PINN predicts a constant ambient temperature that completely misses the subsurface thermal profile, whereas our method accurately captures both the peak magnitude and profile evolution\.

A closer look at Figure[6b](https://arxiv.org/html/2610.09126#S4.F6.sf2)reveals the primary advantages of our approach in capturing heat dissipation regions\. For the materials with low thermal conductivity \(i\.e\., Ti\-6Al\-4V, Inconel 718, and SS 316L\), deviations grow more noticeable over time and at greater distances from the laser track\. Conversely, for AlSi10Mg and Copper with much higher thermal conductivity and thus higher thermal diffusivity \(kcρc​Cp\\frac\{k\_\{c\}\}\{\\rho\_\{c\}C\_\{p\}\}\), the performance gap appears early during laser irradiation on the top surface due to rapid heat dissipation in these highly conductive materials\. As the heat absorbed from the laser irradiation quickly dissipates toward the subsurface due to high thermal diffusivity, the deviations between the two methods are notable at early times rather than at late times, unlike in the in\-distribution materials with low thermal conductivity and low thermal diffusivity\. Therefore, it can be confirmed that our adaptive collocation strategy successfully captures distinct, material\-dependent thermal behaviors, with advantages particularly pronounced in heat\-dissipating regions: at late times in the far field for low\-conductivity materials, and across both early times and near\-to\-far fields for high\-conductivity materials\. Finally, the near\-zero standard deviation of AP\-PINN’s predicted temperature further confirms that this failure stems from a structural limitation in its static collocation\.

Despite the superiority of our method especially in the heat dissipation regions, both approaches exhibit notable deviations from the FEM reference under direct laser irradiation, particularly for low\-conductivity materials\. This discrepancy likely stems in part from the high\-frequency nature of these regions, driven by the steep thermal gradients and spikes induced by laser irradiation\. This aligns with the well\-known spectral bias of neural networks, which is further amplified in PINNs because the residual involves high\-order derivatives, making high\-frequency regions notoriously difficult to resolve\([Tancik et al\., 2020](https://arxiv.org/html/2610.09126#bib.bib26);[Wang et al\., 2021](https://arxiv.org/html/2610.09126#bib.bib25)\)\. Because our problem formulation assumes a uniform temporal distribution of time steps, the sampling density remains uniform near these transient thermal peaks\. Incorporating time steps into our adaptive sampling scheme could alleviate this discrepancy, though at the cost of additional optimization and computational complexity\. A more fundamental solution may lie in alternative feature engineering of the spatiotemporal domain, for instance, mapping to the Fourier domain to better resolve high\-frequency regions \(as in Fourier Neural Operators \(FNO\)\([Li et al\., 2020](https://arxiv.org/html/2610.09126#bib.bib27)\)\), which we leave as a promising direction for future work\.

![Refer to caption](https://arxiv.org/html/2610.09126v1/Fig7a_deviation_fields_3mats_signed_IN718_mix_dense_bc.png)\(a\)Inconel 718 \(kc=11\.4k\_\{c\}=11\.4W m\-1K\-1\)
![Refer to caption](https://arxiv.org/html/2610.09126v1/Fig7b_deviation_fields_3mats_signed_Cu_mix_dense_bc.png)\(b\)Copper \(kc=401k\_\{c\}=401W m\-1K\-1\)

Figure 7:Temperature deviationΔ​T=Tθ−T\\Delta T=T\_\{\\theta\}\-Tatt=0\.5t=0\.5,1\.51\.5and2\.52\.5s \(rows\), with each method’s collocation points overlaid\. Top\-surface panels show the top boundary, cross\-section panels the domain group; the dashed line marks the laser center\. Both the total number of collocation points are identical while AP\-PINN draws them fromb⁡\(x∣t\)b\(x\\mid t\)whereas the proposed method from the mixtureqαbq\_\{\\alpha\}^\{b\}in \([10](https://arxiv.org/html/2610.09126#S2.E10)\) withα=0\.6\\alpha=0\.6\. The collocation points drawn in AP\-PINN are concentrated nearby the laser irradiation zone and the top surface, which results in significant discrepancy from the FEM reference at heat dissipation regions\. In contrast, the proposed method effectively allocates the collocation points on both the laser irradiation zone and the heat dissipation regions at a given time step, resulting in considerable attenuation of the discrepancies shown in the AP\-PINN baseline\.
##### Qualitative Analysis

Figure[7](https://arxiv.org/html/2610.09126#S4.F7)compares the top\-surface and cross\-sectional temperature field deviations relative to the ground\-truth FEM solutions for both the AP\-PINN baseline and our proposed method att=0\.5t=0\.5,1\.51\.5, and2\.52\.5s\. Overlaid on the deviation fields are the collocation points sampled by each strategy: the conventional, manually crafted static sampling and our domain\-informed adaptive sampling\. Here, Inconel 718 and Copper are selected as representative low\- and high\-thermal\-conductivity materials, respectively\. Deviations for the remaining evaluation materials are provided in Appendix[A\.8](https://arxiv.org/html/2610.09126#A1.SS8)\.

Notably, the proposed method substantially alleviates the deviation between the AP\-PINN baseline and FEM reference in all panels\. AP\-PINN exhibits a growing region of under\-prediction in heat dissipation regions for both materials and at every time step\. For the low\-conductivity material Inconel 718, at the early time step \(t=0\.5t=0\.5s\), when the laser has only just begun irradiating the top surface, the absorbed heat remains concentrated near the melt pool, and the temperature deviations are similarly localized to this region\. As the laser moves along the track and time passes, this failure to capture heat dissipation propagates along each material’s heat dissipation path: toward the subsurface where heat accumulates for the low\-conductivity Inconel 718, and nearly parallel to the laser track for the high\-conductivity Copper\. These consistent failures highlight a fundamental limitation of conventional, manually crafted static sampling, which concentrates points almost exclusively near the laser track to ensure stable convergence of PINNs in metal AM, leaving substantial room for improvement in capturing the rapid\-cooling zones where heat dissipates\.

Our domain\-informed adaptive proposal addresses this limitation by augmenting the conventional sampling region with a learned proposal that accounts for the distinct thermal behaviors induced by process conditions, dynamically allocating points along the evolving heat dissipation pathways where errors would otherwise accumulate\. Because these pathways vary considerably with thermophysical material properties, effective collocation requires more than sampling the melt pool along the laser track alone\. By dynamically reallocating points to these fast\-cooling, high\-gradient regions, our method suppresses the deviations prominent in the baseline to within approximately±10\\pm 10K across all time steps and all five evaluation materials, with progressively larger gains as heat continues to accumulate and dissipate over time\.

![Refer to caption](https://arxiv.org/html/2610.09126v1/Fig8_F16f_marginals_per_material_1fm_src3535.png)Figure 8:Marginal distributions of collocation points along each spatial axis, averaged overtt, across the five evaluation materials\.
##### Collocation Sampling Distribution Comparison

Figure[8](https://arxiv.org/html/2610.09126#S4.F8)presents the marginal distributions along each spatial dimension where the conventional approach and our adaptive sampling proposal concentrate collocation points\. The conventional, domain\-informed yet static sampling strategy distributes points uniformly along thexx\- andyy\-dimensions, while along thezz\-dimension it concentrates points predominantly near the top surface with a coarser distribution elsewhere\. In contrast, our proposed method samples more heavily in high\-residual regions, particularly during early time steps and near the boundaries\. This behavior aligns with the findings in Figure[7](https://arxiv.org/html/2610.09126#S4.F7): at an early time step \(t=0\.5t=0\.5s\), when heat from laser irradiation has not yet fully dissipated toward the boundaries, collocation points are concentrated primarily near the melt pool\. As time progresses and the laser traverses the track, heat propagates throughout the domain, and high\-residual regions accumulate in the far\-field subsurface where heat dissipates rather than remaining confined to the vicinity of the laser track\. Accordingly, the points generated by our adaptive proposal shift toward lowerzz\-levels rather than remaining restricted to the top surface\.

Most importantly, the distribution of these collocation points depends strongly on the thermophysical properties of each material, as evidenced by the varying marginal distributions across materials\. As thermal conductivity increases, the marginal distributions along thexx\- andyy\-dimensions become smoother, reflecting the enhanced heat diffusivity that requires allocating more points farther from the laser track\. Similarly, for thezz\-marginal distribution, lower\-diffusivity materials retain points mainly near the top surface \(z=6z=6mm\), while higher thermal diffusivity drives the point concentration deeper toward the bottom, where heat accumulates and dissipates to form high\-residual zones, a trend further corroborated by Figure[7](https://arxiv.org/html/2610.09126#S4.F7)\.

##### Ablation Study and Sensitivity Analysis

To validate our domain\-informed adaptive sampling strategy, we conduct an ablation study and sensitivity analysis on the mixture formulationqαb​\(x∣t,μ\):=\(1−α\)​gϕ​\(x∣t,μ\)\+α​b​\(x∣t\)q\_\{\\alpha\}^\{b\}\(x\\mid t,\\mu\):=\(1\-\\alpha\)\\,g\_\{\\phi\}\(x\\mid t,\\mu\)\+\\alpha\\,b\(x\\mid t\)by varying the mixture weightα∈\[0,1\]\\alpha\\in\[0,1\]and comparing our domain\-informed base distributionb⁡\(x∣t\)b\(x\\mid t\)against an uninformative uniform base\.

![Refer to caption](https://arxiv.org/html/2610.09126v1/Fig9_alpha_sweep_final5_std.png)Figure 9:RelativeL2L\_\{2\}error \(mean±\\pmstd over five seeds\) across the mixture weightα\\alphafor the two base distributions\.As shown in Figure[9](https://arxiv.org/html/2610.09126#S4.F9), the domain\-informed base consistently outperforms the uniform base across nearly all values ofα\\alpha\. The proposed method achieves its lowest relativeL2L\_\{2\}error on a broad plateau overα∈\[0\.5,0\.8\]\\alpha\\in\[0\.5,0\.8\], reaching a minimum atα=0\.6\\alpha=0\.6\(1\.48±0\.31%1\.48\\pm 0\.31\\%\)\. This wide plateau highlights low sensitivity toα\\alpha, which stems from the complementary balance between the two sampling components: the learned proposalgϕg\_\{\\phi\}concentrates on heat\-dissipating regions, while the base distribution ensures coverage of highly transient regions near the laser path\. Conversely, settingα=0\\alpha=0\(relying solely on the learned proposal\) leads to degraded accuracy due to the lack of coverage in the transient regions\.

Crucially, the base distribution governs training stability under the low\-regularity conditions of metal AM\. While the uniform base yields somewhat comparable performance forα≤0\.7\\alpha\\leq 0\.7, its performance degrades rapidly at higherα\\alpha, resulting in frequent convergence failures \(α=0\.9\\alpha=0\.9\) and complete collapse \(α=1\\alpha=1\) due to cold\-starting with uniformly sampled collocation points under steep thermal gradients\. In contrast, the domain\-informed base completely eliminates these bimodal failure modes as in previous research\. This confirms that incorporating domain knowledge via the base distribution is essential to simultaneously achieve peak accuracy and guaranteed training stability\.

![Refer to caption](https://arxiv.org/html/2610.09126v1/Fig10a_deviation_fields_3mats_basecmp_IN718_s4vs5.png)\(a\)Inconel 718
![Refer to caption](https://arxiv.org/html/2610.09126v1/Fig10b_deviation_fields_3mats_basecmp_Cu_s4vs5.png)\(b\)Copper

Figure 10:Signed deviationΔ​T=TPINN−TFEM\\Delta T=T\_\{\\text\{PINN\}\}\-T\_\{\\text\{FEM\}\}att=0\.5t=0\.5,1\.51\.5,2\.52\.5s for models trained with the domain\-informed and the uniform base at the sameα=0\.6\\alpha=0\.6, with each configuration’s collocation points overlaid\.To physically clarify this performance gap atα=0\.6\\alpha=0\.6, Figure[10](https://arxiv.org/html/2610.09126#S4.F10)compares the deviation fields under both base distributions\. The domain\-informed base places high sampling density within1​mm1\\,\\mathrm\{mm\}of the laser track to ensure coverage of highly transient regions, whereas the uniform base severely undersamples this zone, leading to early\-time underestimation near the heat source\. This spatial undersampling manifests according to material properties: in low\-conductivity Inconel 718, errors trail the moving beam along the traversed path, whereas in highly conductive copper \(≈35×\\approx 35\\timeshigher conductivity\), errors travel dynamically beneath the laser\. By embedding domain knowledge to guarantee coverage near steep thermal gradients, the domain\-informed base eliminates these systematic errors, yielding nearly featureless deviation fields across all materials\.

##### Overall discussion

In summary, our experiments show that the proposed adaptive sampling strategy substantially outperforms both baselines, reducing AP\-PINN’s relativeL2L\_\{2\}error by an average of62\.1%62\.1\\%and up to70\.1%70\.1\\%across all five materials\. Despite higher training losses, the method achieves a lower estimated population loss by concentrating collocation points in high\-residual regions, which translates into accurate subsurface and far\-field thermal predictions where AP\-PINN’s static sampling fails\. This adaptivity is material\-dependent, dynamically shifting according to each material’s thermophysical properties, and the ablation study confirms that incorporating metal AM domain knowledge is essential for both predictive accuracy and training stability\.

While the proposed method shows considerable effectiveness and is, to our knowledge, the first adaptive collocation sampling method introduced in the PINN literature for metal AM, several limitations remain\. First, the state\-of\-the\-art baseline and ours still show noticeable deviations near the melt pool under direct laser irradiation due to the extremely high\-frequency thermal gradients\. Second, the necessity of the domain\-informed base distributionb⁡\(x∣t\)b\(x\\mid t\)and the choice of mixture weightα\\alphaare justified primarily through empirical observations rather than a principled theoretical account\. Third, our ground\-truth data are obtained from FEM simulations, a widely adopted numerical benchmark in this literature; nonetheless, validation against actual experiments would further strengthen confidence in the method’s applicability\. Finally, among process conditions we vary only the feedstock material, which aligns with our main objective since thermophysical properties reshape the residual landscape more strongly than process parameters such as laser power or scan speed; still, incorporating such parameters and varying laser scan paths would help establish broader applicability\.

These limitations suggest several directions for future work\. As discussed earlier, incorporating the temporal dimension into the adaptive sampling scheme, or adopting Fourier\-based feature encoding, could better resolve high\-frequency thermal transients near the laser track\. Developing a theoretical account of the domain\-informed base distribution, along with a principled procedure for selectingα\\alpha, would strengthen the method’s foundations\. Validating predicted thermal histories against actual experiments would further test its applicability beyond FEM\-based benchmarks\. Finally, extending the framework to jointly vary process parameters and laser scan paths, in addition to feedstock materials, would help establish broader applicability across the full space of AM process conditions\.

## 5Conclusion

This work presents, to the best of our knowledge, the first adaptive collocation sampling strategy for PINNs in metal AM\. We theoretically show that the generalization error of parametric PINNs is governed by the variance of the empirical loss estimator under the sampling proposal, and that any static,μ\\mu\-independent proposal faces a strict variance ceiling, making it strictly less effective than adaptive sampling that accounts for process conditions\. Guided by this analysis, we adopt Conditional Flow Matching so that a single amortized generator produces the near\-optimal proposal for any process condition in one forward pass, and integrate metal AM domain knowledge by mixing its samples with a domain\-informed base distribution for training stability\.

Extensive experiments on a widely adopted metal AM benchmark demonstrate that the proposed method reduces the relativeL2L\_\{2\}error of the state\-of\-the\-art parametric PINN baseline by an average of62\.1%62\.1\\%and up to70\.1%70\.1\\%, with less than1%1\\%additional training overhead\. Analysis of the empirical and the estimated population loss, thermal history predictions, and qualitative deviation fields further reveals that this improvement stems from the ability of our adaptive proposal to dynamically reallocate collocation points toward heat dissipation regions, which conventional sampling overlooks and therefore fails to capture at increasing depth and distance from the laser path\. The accompanying ablation and sensitivity analysis confirms that the domain\-informed base distribution is essential for training stability, complementing the learned proposal to jointly achieve peak accuracy and robust convergence\.

Taken together, our theoretical and empirical results establish that adaptive, process condition\-aware collocation sampling is critical to realizing the generalization potential of parametric PINNs in metal AM\. Building on the limitations and future directions discussed above, we hope this work encourages a shift from static, manually crafted collocation strategies toward theoretically grounded, adaptive sampling frameworks for PINNs in metal AM and beyond\.

## Appendix AAppendix

### A\.1Nomenclature

SymbolMeaningUnitGeneral notation\(x,t\)\(x,t\)spatiotemporal collocation pointΩ,Ωx,tend\\Omega,\\ \\Omega\_\{x\},\\ t\_\{\\mathrm\{end\}\}spatiotemporal / spatial domain; simulated horizonμ,𝒫\\mu,\\ \\mathcal\{P\}process conditionμ=\(ρc,Cp,kc\)\\mu=\(\\rho\_\{c\},C\_\{p\},k\_\{c\}\); its \(compact\) parameter spacepptarget density \(uniform overΩ×𝒫\\Omega\\times\\mathcal\{P\}\)q,qθ⋆,b,qαbq,\\ q^\{\\star\}\_\{\\theta\},\\ b,\\ q\_\{\\alpha\}^\{b\}general proposal; variance\-optimal proposal; domain\-informed base; mixturegϕg\_\{\\phi\}learned CFM generator approximatingqθsrc⋆q^\{\\star\}\_\{\\theta\_\{\\mathrm\{src\}\}\}α\\alphamixture weightfθf\_\{\\theta\}predictor network with learnable parametersθ\\thetarθ,pde,rθ,bc,rθ,icr\_\{\\theta,\\mathrm\{pde\}\},\\ r\_\{\\theta,\\mathrm\{bc\}\},\\ r\_\{\\theta,\\mathrm\{ic\}\}PDE / BC / IC residualsλpde,λbc,λic,R\\lambda\_\{\\mathrm\{pde\}\},\\ \\lambda\_\{\\mathrm\{bc\}\},\\ \\lambda\_\{\\mathrm\{ic\}\},\\ Rresidual balancing weights; residual boundℒ⁡\(θ\),ℒ^N​\(θ,q\)\\mathcal\{L\}\(\\theta\),\\ \\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)population loss; empirical loss estimatorMq,σq2​\(θ\),σ¯q2M\_\{q\},\\ \\sigma\_\{q\}^\{2\}\(\\theta\),\\ \\bar\{\\sigma\}\_\{q\}^\{2\}likelihood\-ratio bound; estimator variance; its supremum overΘ\\ThetaVc⋆,Vnc⋆V\_\{\\mathrm\{c\}\}^\{\\star\},\\ V\_\{\\mathrm\{nc\}\}^\{\\star\}minimal variance of conditional / unconditional proposal familiesτ,vϕ,zτ,ℓ\\tau,\\ v\_\{\\phi\},\\ z\_\{\\tau\},\\ \\ellflow\-matching time; velocity field; flow state; residual\-group identifierϱθ\\varrho\_\{\\theta\}normalized spatial shape of the squared residual \(Appendix[A\.6](https://arxiv.org/html/2610.09126#A1.SS6)\)N,G,Mres,B,J,NODEN,\\ G,\\ M\_\{\\mathrm\{res\}\},\\ B,\\ J,\\ N\_\{\\mathrm\{ODE\}\}collocation batch; pool size; resample size; CFM batch; stored conditions; Euler stepsKwarm,Kflow,KPINN,θsrcK\_\{\\mathrm\{warm\}\},\\ K\_\{\\mathrm\{flow\}\},\\ K\_\{\\mathrm\{PINN\}\},\\ \\theta\_\{\\mathrm\{src\}\}stage iteration counts; warm\-start \(source\) predictor parametersPhysical quantitiesxxspatial coordinatem\\mathrm\{m\}ttphysical times\\mathrm\{s\}T,TθT,\\ T\_\{\\theta\}ground\-truth / predicted temperature fieldK\\mathrm\{K\}T0T\_\{0\}ambient \(initial\) temperatureK\\mathrm\{K\}ρc\\rho\_\{c\}densitykg/m3\\mathrm\{kg/m^\{3\}\}CpC\_\{p\}specific heat capacityJ/\(kg⋅K\)\\mathrm\{J/\(kg\\cdot K\)\}kck\_\{c\}thermal conductivityW/\(m⋅K\)\\mathrm\{W/\(m\\cdot K\)\}PPlaser powerW\\mathrm\{W\}η\\etalaser absorptivity–RlaserR\_\{\\mathrm\{laser\}\}laser beam radiusm\\mathrm\{m\}xlaser​\(t\),𝐯,vx\_\{\\mathrm\{laser\}\}\(t\),\\ \\mathbf\{v\},\\ vlaser center position; scan velocity; scan speedm\\mathrm\{m\};m/s\\mathrm\{m/s\};m/s\\mathrm\{m/s\}hhconvective heat transfer coefficientW/\(m2⋅K\)\\mathrm\{W/\(m^\{2\}\\cdot K\)\}eesurface emissivity–σ\\sigmaStefan–Boltzmann constantW/\(m2⋅K4\)\\mathrm\{W/\(m^\{2\}\\cdot K^\{4\}\)\}qlaser,qconv,qradq\_\{\\mathrm\{laser\}\},\\ q\_\{\\mathrm\{conv\}\},\\ q\_\{\\mathrm\{rad\}\}laser / convective / radiative heat fluxW/m2\\mathrm\{W/m^\{2\}\}

### A\.2Governing Equations, Boundary Conditions, and Residuals

The spatial domain isΩx⊂ℝdx\\Omega\_\{x\}\\subset\\mathbb\{R\}^\{d\_\{x\}\}withdx=3d\_\{x\}=3, the time interval is\[0,tend\]\[0,t\_\{\\mathrm\{end\}\}\], andΩ=Ωx×\[0,tend\]\\Omega=\\Omega\_\{x\}\\times\[0,t\_\{\\mathrm\{end\}\}\]\. The PDE parameter vector collects the three thermophysical material properties that are varied in our experimentsμ=\(ρc,Cp,kc\)∈𝒫⊂ℝm\\mu=\(\\rho\_\{c\},\\,C\_\{p\},\\,k\_\{c\}\)\\in\\mathcal\{P\}\\subset\\mathbb\{R\}^\{m\}, namely the densityρc\\rho\_\{c\}, the specific heat capacityCpC\_\{p\}, and the thermal conductivitykck\_\{c\}\. All remaining quantities appearing below, including the process parameters and the surface exchange coefficients, are held fixed\.

##### Governing PDE\.

Transient heat transfer in the bare\-plate metal AM process is governed by conservation of energy coupled with Fourier’s law of heat conduction\. The exact temperature fieldT⁡\(x,t,μ\)T\(x,t;\\mu\)satisfies

𝒩\[T\]\(x,t;μ\):=ρcCp∂T∂t−∇⋅\(kc∇T\)=0,\(x,t\)∈Ωx×\(0,tend\],\\mathcal\{N\}\[T\]\(x,t;\\mu\):=\\rho\_\{c\}\\,C\_\{p\}\\,\\frac\{\\partial T\}\{\\partial t\}\-\\nabla\\\!\\cdot\\\!\\bigl\(k\_\{c\}\\,\\nabla T\\bigr\)=0,\\qquad\(x,t\)\\in\\Omega\_\{x\}\\times\(0,t\_\{\\mathrm\{end\}\}\],\(11\)where𝒩⁡\[⋅\]\\mathcal\{N\}\[\\cdot\]is the nonlinear differential operator introduced in Sec\.[2\.2](https://arxiv.org/html/2610.09126#S2.SS2); the nonlinearity enters through the radiative boundary term below\.

##### Boundary conditions\.

The boundary∂Ωx\\partial\\Omega\_\{x\}is decomposed into the top surfaceΓtop\\Gamma\_\{\\mathrm\{top\}\}, the side surfacesΓside\\Gamma\_\{\\mathrm\{side\}\}, and the bottom surfaceΓbottom\\Gamma\_\{\\mathrm\{bottom\}\}, so that∂Ωx=Γtop∪Γside∪Γbottom\\partial\\Omega\_\{x\}=\\Gamma\_\{\\mathrm\{top\}\}\\cup\\Gamma\_\{\\mathrm\{side\}\}\\cup\\Gamma\_\{\\mathrm\{bottom\}\}\. With𝐧\\mathbf\{n\}the outward unit normal, the boundary operatorℬ⁡\[⋅\]\\mathcal\{B\}\[\\cdot\]is defined piecewise as

ℬ⁡\[T\]​\(x,t,μ\):=\{−kc∇T⋅𝐧−\(qlaser\+qconv\+qrad\),x∈Γtop,−kc∇T⋅𝐧−\(qconv\+qrad\),x∈Γside,T⁡\(x,t,μ\)−T0,x∈Γbottom,\\mathcal\{B\}\[T\]\(x,t;\\mu\):=\\begin\{cases\}\-k\_\{c\}\\,\\nabla T\\cdot\\mathbf\{n\}\-\\bigl\(q\_\{\\mathrm\{laser\}\}\+q\_\{\\mathrm\{conv\}\}\+q\_\{\\mathrm\{rad\}\}\\bigr\),&x\\in\\Gamma\_\{\\mathrm\{top\}\},\\\\\[4\.0pt\] \-k\_\{c\}\\,\\nabla T\\cdot\\mathbf\{n\}\-\\bigl\(q\_\{\\mathrm\{conv\}\}\+q\_\{\\mathrm\{rad\}\}\\bigr\),&x\\in\\Gamma\_\{\\mathrm\{side\}\},\\\\\[4\.0pt\] T\(x,t;\\mu\)\-T\_\{0\},&x\\in\\Gamma\_\{\\mathrm\{bottom\}\},\\end\{cases\}\(12\)fort∈\(0,tend\]t\\in\(0,t\_\{\\mathrm\{end\}\}\], and the boundary conditions readℬ​\[T\]​\(x,t,μ\)=0\\mathcal\{B\}\[T\]\(x,t;\\mu\)=0on∂Ωx×\(0,tend\]\\partial\\Omega\_\{x\}\\times\(0,t\_\{\\mathrm\{end\}\}\]\.

The laser heat flux imposed on the top surface is modeled as a Gaussian surface heat source,

qlaser​\(x,t\)=2​η​Pπ​Rlaser2​exp⁡\(−2​∥x−xlaser​\(t\)∥2Rlaser2\),q\_\{\\mathrm\{laser\}\}\(x,t\)=\\frac\{2\\,\\eta\\,P\}\{\\pi R\_\{\\mathrm\{laser\}\}^\{2\}\}\\exp\\\!\\left\(\-\\frac\{2\\,\\lVert x\-x\_\{\\mathrm\{laser\}\}\(t\)\\rVert^\{2\}\}\{R\_\{\\mathrm\{laser\}\}^\{2\}\}\\right\),\(13\)whereη\\etais the laser absorptivity,PPthe laser power, andRlaserR\_\{\\mathrm\{laser\}\}the laser beam radius\. The laser center follows the prescribed scan pathxlaser​\(t\)=x0\+𝐯​tx\_\{\\mathrm\{laser\}\}\(t\)=x\_\{0\}\+\\mathbf\{v\}\\,t; a one\-dimensional scan along thexx\-axis on the top surface is considered, i\.e\.,𝐯=\(v,0,0\)\\mathbf\{v\}=\(v,0,0\)\. Convective and radiative fluxes act on all surfaces exceptΓbottom\\Gamma\_\{\\mathrm\{bottom\}\}, where a fixed\-temperature Dirichlet condition is imposed:

qconv​\(x,t,μ\)\\displaystyle q\_\{\\mathrm\{conv\}\}\(x,t;\\mu\)=h⁡\(T⁡\(x,t,μ\)−T0\),\\displaystyle=h\\bigl\(T\(x,t;\\mu\)\-T\_\{0\}\\bigr\),\(14\)qrad​\(x,t,μ\)\\displaystyle q\_\{\\mathrm\{rad\}\}\(x,t;\\mu\)=σ​e​\(T​\(x,t,μ\)4−T04\),\\displaystyle=\\sigma\\,e\\bigl\(T\(x,t;\\mu\)^\{4\}\-T\_\{0\}^\{4\}\\bigr\),\(15\)withhhthe convective heat transfer coefficient,σ\\sigmathe Stefan–Boltzmann constant, andeethe surface emissivity\. Note thatqlaserq\_\{\\mathrm\{laser\}\}is independent ofμ\\mu, whereasqconvq\_\{\\mathrm\{conv\}\}andqradq\_\{\\mathrm\{rad\}\}inherit the parameter dependence of the temperature field\. The quartic term in \([15](https://arxiv.org/html/2610.09126#A1.E15)\) is the source of the nonlinearity of the boundary value problem\.

##### Initial condition\.

ℐ⁡\[T\]​\(x,μ\):=T⁡\(x,0,μ\)−T0=0,x∈Ωx,\\mathcal\{I\}\[T\]\(x;\\mu\):=T\(x,0;\\mu\)\-T\_\{0\}=0,\\qquad x\\in\\Omega\_\{x\},\(16\)withT0T\_\{0\}the ambient temperature\.

##### Residuals\.

Substituting the network approximationTθ​\(x,t,μ\)T\_\{\\theta\}\(x,t;\\mu\)into \([11](https://arxiv.org/html/2610.09126#A1.E11)\)–\([12](https://arxiv.org/html/2610.09126#A1.E12)\) yields the three residual terms appearing in Eq\. \([1](https://arxiv.org/html/2610.09126#S2.E1)\):

rθ,pde​\(x,t,μ\)\\displaystyle r\_\{\\theta,\\mathrm\{pde\}\}\(x,t;\\mu\):=𝒩\[Tθ\]\(x,t;μ\)=ρcCp∂Tθ∂t−∇⋅\(kc∇Tθ\),\\displaystyle:=\\mathcal\{N\}\[T\_\{\\theta\}\]\(x,t;\\mu\)=\\rho\_\{c\}C\_\{p\}\\frac\{\\partial T\_\{\\theta\}\}\{\\partial t\}\-\\nabla\\\!\\cdot\\\!\\bigl\(k\_\{c\}\\nabla T\_\{\\theta\}\\bigr\),\(x,t\)∈Ωx×\(0,tend\],\\displaystyle\(x,t\)\\in\\Omega\_\{x\}\\times\(0,t\_\{\\mathrm\{end\}\}\],\(17\)rθ,bc​\(x,t,μ\)\\displaystyle r\_\{\\theta,\\mathrm\{bc\}\}\(x,t;\\mu\):=ℬ⁡\[Tθ\]​\(x,t,μ\),\\displaystyle:=\\mathcal\{B\}\[T\_\{\\theta\}\]\(x,t;\\mu\),\(x,t\)∈∂Ωx×\(0,tend\],\\displaystyle\(x,t\)\\in\\partial\\Omega\_\{x\}\\times\(0,t\_\{\\mathrm\{end\}\}\],\(18\)rθ,ic​\(x,μ\)\\displaystyle r\_\{\\theta,\\mathrm\{ic\}\}\(x;\\mu\):=ℐ⁡\[Tθ\]​\(x,μ\)=Tθ​\(x,0,μ\)−T0,\\displaystyle:=\\mathcal\{I\}\[T\_\{\\theta\}\]\(x;\\mu\)=T\_\{\\theta\}\(x,0;\\mu\)\-T\_\{0\},x∈Ωx\.\\displaystyle x\\in\\Omega\_\{x\}\.\(19\)Explicitly, the boundary residualrθ,bcr\_\{\\theta,\\mathrm\{bc\}\}combines a Neumann\-type residual onΓtop∪Γside\\Gamma\_\{\\mathrm\{top\}\}\\cup\\Gamma\_\{\\mathrm\{side\}\}and a Dirichlet\-type residual onΓbottom\\Gamma\_\{\\mathrm\{bottom\}\}:

rθ,bc=\{−kc∇Tθ⋅𝐧−qlaser−qconv\[Tθ\]−qrad\[Tθ\],x∈Γtop,−kc∇Tθ⋅𝐧−qconv\[Tθ\]−qrad\[Tθ\],x∈Γside,Tθ​\(x,t,μ\)−T0,x∈Γbottom\.r\_\{\\theta,\\mathrm\{bc\}\}=\\begin\{cases\}\-k\_\{c\}\\,\\nabla T\_\{\\theta\}\\cdot\\mathbf\{n\}\-q\_\{\\mathrm\{laser\}\}\-q\_\{\\mathrm\{conv\}\}\[T\_\{\\theta\}\]\-q\_\{\\mathrm\{rad\}\}\[T\_\{\\theta\}\],&x\\in\\Gamma\_\{\\mathrm\{top\}\},\\\\\[3\.0pt\] \-k\_\{c\}\\,\\nabla T\_\{\\theta\}\\cdot\\mathbf\{n\}\-q\_\{\\mathrm\{conv\}\}\[T\_\{\\theta\}\]\-q\_\{\\mathrm\{rad\}\}\[T\_\{\\theta\}\],&x\\in\\Gamma\_\{\\mathrm\{side\}\},\\\\\[3\.0pt\] T\_\{\\theta\}\(x,t;\\mu\)\-T\_\{0\},&x\\in\\Gamma\_\{\\mathrm\{bottom\}\}\.\\end\{cases\}\(20\)Each residual is evaluated on collocation points drawn from the corresponding subdomain, and the three terms are combined with the balancing weightsλpde,λbc,λic\\lambda\_\{\\mathrm\{pde\}\},\\lambda\_\{\\mathrm\{bc\}\},\\lambda\_\{\\mathrm\{ic\}\}as in Eq\. \([1](https://arxiv.org/html/2610.09126#S2.E1)\)\. Note thatrθ,icr\_\{\\theta,\\mathrm\{ic\}\}is independent ofttsince it is evaluated att=0t=0, and it depends onμ\\muonly throughTθT\_\{\\theta\}itself\.

### A\.3Proof of Excess Generalization Error Decomposition Eq\. \([4](https://arxiv.org/html/2610.09126#S2.E4)\)

We derive the decomposition of the excess generalization error given in \([4](https://arxiv.org/html/2610.09126#S2.E4)\) based on the triangular inequality in statistical learning theory\. Recall thatθ⋆∈arg⁡minθ∈Θ⁡ℒ⁡\(θ\)\\theta^\{\\star\}\\in\\arg\\min\_\{\\theta\\in\\Theta\}\\mathcal\{L\}\(\\theta\)is the population loss minimizer,θ^N​\(q\)∈arg⁡minθ∈Θ​ℒ^N​\(θ,q\)\\widehat\{\\theta\}\_\{N\}\(q\)\\in\\arg\\min\_\{\\theta\\in\\Theta\}\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)is the empirical loss minimizer under proposalqq, andθ~N\\widetilde\{\\theta\}\_\{N\}is the parameter returned after training\. The excess generalization error is decomposed as follows:

ℒ⁡\(θ~N\)−ℒ⁡\(θ⋆\)⏟excess generalization error\\displaystyle\\underbrace\{\\mathcal\{L\}\(\\widetilde\{\\theta\}\_\{N\}\)\-\\mathcal\{L\}\(\\theta^\{\\star\}\)\}\_\{\\text\{excess generalization error\}\}=\[ℒ⁡\(θ~N\)−ℒ^N​\(θ~N,q\)\]\+\[ℒ^N​\(θ~N,q\)−ℒ^N​\(θ^N​\(q\),q\)\]\\displaystyle=\\big\[\\mathcal\{L\}\(\\widetilde\{\\theta\}\_\{N\}\)\-\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\widetilde\{\\theta\}\_\{N\};q\)\\big\]\+\\big\[\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\widetilde\{\\theta\}\_\{N\};q\)\-\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\widehat\{\\theta\}\_\{N\}\(q\);q\)\\big\]\+\[ℒ^N​\(θ^N​\(q\),q\)−ℒ^N​\(θ⋆,q\)\]\+\[ℒ^N​\(θ⋆,q\)−ℒ⁡\(θ⋆\)\]\.\\displaystyle\\quad\+\\big\[\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\widehat\{\\theta\}\_\{N\}\(q\);q\)\-\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta^\{\\star\};q\)\\big\]\+\\big\[\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta^\{\\star\};q\)\-\\mathcal\{L\}\(\\theta^\{\\star\}\)\\big\]\.\(21\)
Sinceθ^N​\(q\)\\widehat\{\\theta\}\_\{N\}\(q\)is the minimizer ofℒ^N​\(⋅,q\)\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\cdot;q\)overΘ\\Theta, andθ⋆∈Θ\\theta^\{\\star\}\\in\\Theta, we haveℒ^N​\(θ^N​\(q\),q\)≤ℒ^N​\(θ⋆,q\)\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\widehat\{\\theta\}\_\{N\}\(q\);q\)\\leq\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta^\{\\star\};q\), which implies:

ℒ^N​\(θ^N​\(q\),q\)−ℒ^N​\(θ⋆,q\)≤0\.\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\widehat\{\\theta\}\_\{N\}\(q\);q\)\-\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta^\{\\star\};q\)\\;\\leq\\;0\.\(22\)
Applying this inequality to \([21](https://arxiv.org/html/2610.09126#A1.E21)\), we obtain:

ℒ⁡\(θ~N\)−ℒ⁡\(θ⋆\)⏟excess generalization error\\displaystyle\\underbrace\{\\mathcal\{L\}\(\\widetilde\{\\theta\}\_\{N\}\)\-\\mathcal\{L\}\(\\theta^\{\\star\}\)\}\_\{\\text\{excess generalization error\}\}≤\[ℒ^N​\(θ~N,q\)−ℒ^N​\(θ^N​\(q\),q\)\]\+\[ℒ⁡\(θ~N\)−ℒ^N​\(θ~N,q\)\]\+\[ℒ^N​\(θ⋆,q\)−ℒ⁡\(θ⋆\)\]\.\\displaystyle\\leq\\big\[\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\widetilde\{\\theta\}\_\{N\};q\)\-\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\widehat\{\\theta\}\_\{N\}\(q\);q\)\\big\]\+\\big\[\\mathcal\{L\}\(\\widetilde\{\\theta\}\_\{N\}\)\-\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\widetilde\{\\theta\}\_\{N\};q\)\\big\]\+\\big\[\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta^\{\\star\};q\)\-\\mathcal\{L\}\(\\theta^\{\\star\}\)\\big\]\.\(23\)
Finally, bounding the last two terms by their supremum over the hypothesis classΘ\\Theta, we get:

ℒ⁡\(θ~N\)−ℒ⁡\(θ⋆\)⏟excess generalization error\\displaystyle\\underbrace\{\\mathcal\{L\}\(\\widetilde\{\\theta\}\_\{N\}\)\-\\mathcal\{L\}\(\\theta^\{\\star\}\)\}\_\{\\text\{excess generalization error\}\}≤\[ℒ^N​\(θ~N,q\)−ℒ^N​\(θ^N​\(q\),q\)\]\+supθ∈Θ\|ℒ^N​\(θ,q\)−ℒ⁡\(θ\)\|\+supθ∈Θ\|ℒ^N​\(θ,q\)−ℒ⁡\(θ\)\|\\displaystyle\\leq\\big\[\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\widetilde\{\\theta\}\_\{N\};q\)\-\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\widehat\{\\theta\}\_\{N\}\(q\);q\)\\big\]\+\\sup\_\{\\theta\\in\\Theta\}\\big\|\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)\-\\mathcal\{L\}\(\\theta\)\\big\|\+\\sup\_\{\\theta\\in\\Theta\}\\big\|\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)\-\\mathcal\{L\}\(\\theta\)\\big\|=\[ℒ^N​\(θ~N,q\)−ℒ^N​\(θ^N​\(q\),q\)\]⏟optimization error\+2​supθ∈Θ\|ℒ^N​\(θ,q\)−ℒ⁡\(θ\)\|⏟statistical error\\displaystyle=\\underbrace\{\\big\[\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\widetilde\{\\theta\}\_\{N\};q\)\-\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\widehat\{\\theta\}\_\{N\}\(q\);q\)\\big\]\}\_\{\\text\{optimization error\}\}\+2\\,\\sup\_\{\\theta\\in\\Theta\}\\underbrace\{\\big\|\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)\-\\mathcal\{L\}\(\\theta\)\\big\|\}\_\{\\text\{statistical error\}\}\(24\)∎

### A\.4Proof of Proposition[2\.1](https://arxiv.org/html/2610.09126#S2.Thmproposition1)

Prior to the proofs hereafter, we summarize the structural assumptions used throughout our analysis:

###### Assumption A\.1\(Regularity\)\.

The target densityppis a probability density onΩ×𝒫\\Omega\\times\\mathcal\{P\}and strictly positive almost everywhere on its support\.

###### Assumption A\.2\(Residual Boundedness\)\.

There exists a constantR<∞R<\\inftysuch thatsupθ∈Θ\|rθ​\(x,t,μ\)\|≤R\\sup\_\{\\theta\\in\\Theta\}\|r\_\{\\theta\}\(x,t;\\mu\)\|\\leq Ralmost everywhere onΩ×𝒫\\Omega\\times\\mathcal\{P\}\.

###### Assumption A\.3\(Proposal Compatibility\)\.

The proposal densityqqsatisfiessupp⁡\(p\)⊆supp⁡\(q\)\\mathrm\{supp\}\(p\)\\subseteq\\mathrm\{supp\}\(q\), and the likelihood ratio is bounded almost everywhere, i\.e\.,Mq:=ess​supq​p⁡\(x,t,μ\)q⁡\(x,t,μ\)=ess​supq​p⁡\(x\)q⁡\(x∣t,μ\)<∞M\_\{q\}:=\\mathrm\{ess\\,sup\}\_\{q\}\\frac\{p\(x,t,\\mu\)\}\{q\(x,t,\\mu\)\}=\\mathrm\{ess\\,sup\}\_\{q\}\\frac\{p\(x\)\}\{q\(x\\mid t,\\mu\)\}<\\infty\.

Building on these assumptions, we state and prove the lemma regarding the variance of the empirical loss estimator:

###### Lemma A\.1\(Variance of the empirical loss\)\.

Under Assumptions[A\.1](https://arxiv.org/html/2610.09126#A1.Thmassumption1)–[A\.3](https://arxiv.org/html/2610.09126#A1.Thmassumption3), for everyθ∈Θ\\theta\\in\\Theta, the variance of the empirical loss estimatorℒ^N​\(θ,q\)\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)satisfies:

Varq​\[ℒ^N​\(θ,q\)\]=1N​σq2​\(θ\),σq2​\(θ\):=𝔼q​\[\(p⁡\(x\)q⁡\(x∣t,μ\)​rθ​\(x,t,μ\)2\)2\]−ℒ​\(θ\)2\.\\mathrm\{Var\}\_\{q\}\\\!\\left\[\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)\\right\]=\\frac\{1\}\{N\}\\,\\sigma\_\{q\}^\{2\}\(\\theta\),\\qquad\\sigma\_\{q\}^\{2\}\(\\theta\):=\\mathbb\{E\}\_\{q\}\\\!\\left\[\\left\(\\frac\{p\(x\)\}\{q\(x\\mid t,\\mu\)\}\\,r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\)^\{\\\!2\}\\,\\right\]\-\\mathcal\{L\}\(\\theta\)^\{2\}\.\(25\)

###### Proof of Lemma[A\.1](https://arxiv.org/html/2610.09126#A1.Thmlemma1)\.

Sinceℒ^N​\(θ,q\)=1N​∑i=1Np⁡\(xi\)q⁡\(xi∣ti,μi\)​rθ​\(xi,ti,μi\)2\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\frac\{p\(x\_\{i\}\)\}\{q\(x\_\{i\}\\mid t\_\{i\},\\mu\_\{i\}\)\}r\_\{\\theta\}\(x\_\{i\},t\_\{i\};\\mu\_\{i\}\)^\{2\}is an average ofNNi\.i\.d\. random variables, its variance is1N\\frac\{1\}\{N\}times the single\-sample variance:

σq2​\(θ\)\\displaystyle\\sigma\_\{q\}^\{2\}\(\\theta\)=Varq​\[p⁡\(x\)q⁡\(x∣t,μ\)​rθ​\(x,t,μ\)2\]\\displaystyle=\\mathrm\{Var\}\_\{q\}\\left\[\\frac\{p\(x\)\}\{q\(x\\mid t,\\mu\)\}\\,r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\]=𝔼q​\[\(p⁡\(x\)q⁡\(x∣t,μ\)​rθ​\(x,t,μ\)2\)2\]−\(𝔼q​\[p⁡\(x\)q⁡\(x∣t,μ\)​rθ​\(x,t,μ\)2\]\)2\.\\displaystyle=\\mathbb\{E\}\_\{q\}\\\!\\left\[\\left\(\\frac\{p\(x\)\}\{q\(x\\mid t,\\mu\)\}\\,r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\)^\{\\\!2\}\\,\\right\]\-\\left\(\\mathbb\{E\}\_\{q\}\\\!\\left\[\\frac\{p\(x\)\}\{q\(x\\mid t,\\mu\)\}\\,r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\]\\right\)^\{2\}\.\(26\)
Expanding the first\-moment expectation inside the second term gives:

𝔼q​\[p⁡\(x\)q⁡\(x∣t,μ\)​rθ​\(x,t,μ\)2\]\\displaystyle\\mathbb\{E\}\_\{q\}\\\!\\left\[\\frac\{p\(x\)\}\{q\(x\\mid t,\\mu\)\}\\,r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\]=∫𝒫∫0tend∫Ωxp⁡\(x\)q⁡\(x∣t,μ\)​rθ​\(x,t,μ\)2⋅\(q⁡\(x∣t,μ\)​p​\(t\)​p​\(μ\)\)​𝑑x​𝑑t​𝑑μ\\displaystyle=\\int\_\{\\mathcal\{P\}\}\\int\_\{0\}^\{t\_\{\\mathrm\{end\}\}\}\\int\_\{\\Omega\_\{x\}\}\\frac\{p\(x\)\}\{q\(x\\mid t,\\mu\)\}\\,r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\cdot\\Big\(q\(x\\mid t,\\mu\)p\(t\)p\(\\mu\)\\Big\)\\,dx\\,dt\\,d\\mu=∫𝒫∫0tend∫Ωxp⁡\(x\)​p​\(t\)​p​\(μ\)⋅rθ​\(x,t,μ\)2​𝑑x​𝑑t​𝑑μ\\displaystyle=\\int\_\{\\mathcal\{P\}\}\\int\_\{0\}^\{t\_\{\\mathrm\{end\}\}\}\\int\_\{\\Omega\_\{x\}\}p\(x\)\\,p\(t\)\\,p\(\\mu\)\\cdot r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\,dx\\,dt\\,d\\mu=𝔼p​\[rθ​\(x,t,μ\)2\]=ℒ⁡\(θ\)\.\\displaystyle=\\mathbb\{E\}\_\{p\}\\\!\\left\[r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\]=\\mathcal\{L\}\(\\theta\)\.\(27\)
Plugging this back into the variance expression, the second term simplifies toℒ​\(θ\)2\\mathcal\{L\}\(\\theta\)^\{2\}, which yields:

σq2​\(θ\)=𝔼q​\[\(p⁡\(x\)q⁡\(x∣t,μ\)​rθ​\(x,t,μ\)2\)2\]−ℒ​\(θ\)2\.\\sigma\_\{q\}^\{2\}\(\\theta\)=\\mathbb\{E\}\_\{q\}\\\!\\left\[\\left\(\\frac\{p\(x\)\}\{q\(x\\mid t,\\mu\)\}\\,r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\)^\{\\\!2\}\\,\\right\]\-\\mathcal\{L\}\(\\theta\)^\{2\}\.\(28\)Multiplying by1N\\frac\{1\}\{N\}completes the proof of Lemma[A\.1](https://arxiv.org/html/2610.09126#A1.Thmlemma1)\. ∎

###### Proof of Proposition[2\.1](https://arxiv.org/html/2610.09126#S2.Thmproposition1)\.

The proof proceeds in three intuitive steps: establishing a pointwise concentration bound for a fixedθ\\theta, uniformizing the bound over the parameter spaceΘ\\Theta, and isolating the precise dependence on the proposal distributionqq\.

##### Step 1: Pointwise Concentration via Bernstein’s Inequality\.

Fix any parameter vectorθ∈Θ\\theta\\in\\Theta\. For theii\-th collocation point\(xi,ti,μi\)∼q\(x\_\{i\},t\_\{i\},\\mu\_\{i\}\)\\sim q, define the importance\-weighted random variable:

Zi​\(θ\):=p⁡\(xi,ti,μi\)q⁡\(xi,ti,μi\)​rθ​\(xi,ti,μi\)2=p⁡\(xi\)q⁡\(xi∣ti,μi\)​rθ​\(xi,ti,μi\)2\.Z\_\{i\}\(\\theta\):=\\frac\{p\(x\_\{i\},t\_\{i\},\\mu\_\{i\}\)\}\{q\(x\_\{i\},t\_\{i\},\\mu\_\{i\}\)\}\\,r\_\{\\theta\}\(x\_\{i\},t\_\{i\};\\mu\_\{i\}\)^\{2\}=\\frac\{p\(x\_\{i\}\)\}\{q\(x\_\{i\}\\mid t\_\{i\},\\mu\_\{i\}\)\}\\,r\_\{\\theta\}\(x\_\{i\},t\_\{i\};\\mu\_\{i\}\)^\{2\}\.\(29\)By definition, the Monte Carlo estimatorℒ^N​\(θ,q\)=1N​∑i=1NZi​\(θ\)\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}Z\_\{i\}\(\\theta\)is an unbiased estimator of the population loss, i\.e\.,𝔼q​\[Zi​\(θ\)\]=ℒ⁡\(θ\)\\mathbb\{E\}\_\{q\}\[Z\_\{i\}\(\\theta\)\]=\\mathcal\{L\}\(\\theta\)\.

We characterize two essential moment properties ofZi​\(θ\)Z\_\{i\}\(\\theta\):

1. \(i\)Per\-sample Variance:Varq​\(Zi​\(θ\)\)=σq2​\(θ\)\\mathrm\{Var\}\_\{q\}\\big\(Z\_\{i\}\(\\theta\)\\big\)=\\sigma\_\{q\}^\{2\}\(\\theta\), as established in Lemma[A\.1](https://arxiv.org/html/2610.09126#A1.Thmlemma1)\.
2. \(ii\)Essential Boundedness:By Assumptions[A\.2](https://arxiv.org/html/2610.09126#A1.Thmassumption2)and[A\.3](https://arxiv.org/html/2610.09126#A1.Thmassumption3), 0≤Zi​\(θ\)≤ess​supq​\(p⁡\(x\)q⁡\(x∣t,μ\)\)⋅sup\(x,t,μ\)rθ​\(x,t,μ\)2≤Mq​R2=:Uq\.0\\leq Z\_\{i\}\(\\theta\)\\leq\\mathrm\{ess\\,sup\}\_\{q\}\\left\(\\frac\{p\(x\)\}\{q\(x\\mid t,\\mu\)\}\\right\)\\cdot\\sup\_\{\(x,t,\\mu\)\}r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\leq M\_\{q\}R^\{2\}=:U\_\{q\}\.\(30\)

Applying Bernstein’s inequality to the i\.i\.d\. sequence\{Zi​\(θ\)−ℒ⁡\(θ\)\}i=1N\\\{Z\_\{i\}\(\\theta\)\-\\mathcal\{L\}\(\\theta\)\\\}\_\{i=1\}^\{N\}, we obtain for anyϵ\>0\\epsilon\>0:

ℙq​\(\|ℒ^N​\(θ,q\)−ℒ⁡\(θ\)\|\>ϵ\)≤2​exp⁡\(−N​ϵ22​σq2​\(θ\)\+23​Uq​ϵ\)\.\\mathbb\{P\}\_\{q\}\\Big\(\\big\|\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)\-\\mathcal\{L\}\(\\theta\)\\big\|\>\\epsilon\\Big\)\\;\\leq\\;2\\exp\\left\(\-\\frac\{N\\epsilon^\{2\}\}\{2\\sigma\_\{q\}^\{2\}\(\\theta\)\+\\frac\{2\}\{3\}U\_\{q\}\\epsilon\}\\right\)\.\(31\)

##### Step 2: Uniform Generalization overΘ\\Theta\.

To extend this concentration uniformly over allθ∈Θ\\theta\\in\\Theta, we control the capacity of the hypothesis function classℱΘ:=\{\(x,t,μ\)↦p⁡\(x\)q⁡\(x∣t,μ\)​rθ​\(x,t,μ\)2:θ∈Θ\}\\mathcal\{F\}\_\{\\Theta\}:=\\\{\(x,t,\\mu\)\\mapsto\\frac\{p\(x\)\}\{q\(x\\mid t,\\mu\)\}r\_\{\\theta\}\(x,t;\\mu\)^\{2\}:\\theta\\in\\Theta\\\}\.

By standard uniform concentration techniques, for any specified confidence parameterδ∈\(0,1\)\\delta\\in\(0,1\), there exist a function class complexity metric𝒞⁡\(Θ,δ\)\>0\\mathcal\{C\}\(\\Theta,\\delta\)\>0and universal numerical constantsC1,C2\>0C\_\{1\},C\_\{2\}\>0such that, with probability at least1−δ1\-\\delta:

supθ∈Θ\|ℒ^N​\(θ,q\)−ℒ⁡\(θ\)\|≤C1​σ¯q2​𝒞​\(Θ,δ\)N\+C2​Mq​R2​𝒞​\(Θ,δ\)N,\\sup\_\{\\theta\\in\\Theta\}\\big\|\\widehat\{\\mathcal\{L\}\}\_\{N\}\(\\theta;q\)\-\\mathcal\{L\}\(\\theta\)\\big\|\\;\\leq\\;C\_\{1\}\\sqrt\{\\frac\{\\bar\{\\sigma\}\_\{q\}^\{2\}\\,\\mathcal\{C\}\(\\Theta,\\delta\)\}\{N\}\}\\;\+\\;C\_\{2\}\\frac\{M\_\{q\}R^\{2\}\\,\\mathcal\{C\}\(\\Theta,\\delta\)\}\{N\},\(32\)whereσ¯q2:=supθ∈Θσq2​\(θ\)\\bar\{\\sigma\}\_\{q\}^\{2\}:=\\sup\_\{\\theta\\in\\Theta\}\\sigma\_\{q\}^\{2\}\(\\theta\)represents the worst\-case per\-sample variance acrossΘ\\Theta\.

##### Step 3: Isolating the Dependence on Proposalqq\.

Note that the complexity term𝒞⁡\(Θ,δ\)\\mathcal\{C\}\(\\Theta,\\delta\)is evaluated under the intrinsic target measurepp, rendering𝒞⁡\(Θ,δ\)\\mathcal\{C\}\(\\Theta,\\delta\)and the constantsC1,C2C\_\{1\},C\_\{2\}completely independent ofqq\.

The choice of proposal distributionqqaffects the upper bound strictly through:

1. 1\.The leading variance termσ¯q2\\bar\{\\sigma\}\_\{q\}^\{2\},
2. 2\.The secondary likelihood ratio boundMqM\_\{q\}\.

In practical training regimes where the sample sizeNNsatisfiesN≳Mq​R2​𝒞σ¯q2N\\gtrsim\\frac\{M\_\{q\}R^\{2\}\\mathcal\{C\}\}\{\\bar\{\\sigma\}\_\{q\}^\{2\}\}, the𝒪⁡\(1/N\)\\mathcal\{O\}\(1/\\sqrt\{N\}\)term dominates the𝒪⁡\(1/N\)\\mathcal\{O\}\(1/N\)term\. Thus, minimizing the estimator varianceσ¯q2\\bar\{\\sigma\}\_\{q\}^\{2\}tightens the leading term of the generalization bound, which completes the proof\. ∎

### A\.5Proof of Theorem[2\.1](https://arxiv.org/html/2610.09126#S2.Thmtheorem1)\.

We recall from \([28](https://arxiv.org/html/2610.09126#A1.E28)\) that sinceℒ​\(θ\)2\\mathcal\{L\}\(\\theta\)^\{2\}does not depend on the proposal choiceqq, minimizingσq2​\(θ\)\\sigma\_\{q\}^\{2\}\(\\theta\)is equivalent to minimizing the second\-moment term𝔼q​\[\(p⁡\(x\)q⁡\(x∣t,μ\)​rθ​\(x,t,μ\)2\)2\]\\mathbb\{E\}\_\{q\}\\\!\\left\[\\left\(\\frac\{p\(x\)\}\{q\(x\\mid t,\\mu\)\}\\,r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\)^\{\\\!2\}\\right\]\.

##### Case 1: Conditional Spatial Proposal \(q∈𝒬cq\\in\\mathcal\{Q\}\_\{c\}\)

Forq∈𝒬cq\\in\\mathcal\{Q\}\_\{c\}, the spatial proposalq⁡\(x∣t,μ\)q\(x\\mid t,\\mu\)depends on both timettand parameterμ\\mu\. Expanding the second moment explicitly:

𝔼q​\[\(p⁡\(x\)q⁡\(x∣t,μ\)​rθ​\(x,t,μ\)2\)2\]\\displaystyle\\mathbb\{E\}\_\{q\}\\\!\\left\[\\left\(\\frac\{p\(x\)\}\{q\(x\\mid t,\\mu\)\}\\,r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\)^\{\\\!2\}\\,\\right\]=∫𝒫p⁡\(μ\)​∫0tendp⁡\(t\)​∫Ωxq⁡\(x∣t,μ\)⋅p​\(x\)2q​\(x∣t,μ\)2​rθ​\(x,t,μ\)4​𝑑x​𝑑t​𝑑μ\\displaystyle=\\int\_\{\\mathcal\{P\}\}p\(\\mu\)\\int\_\{0\}^\{t\_\{\\mathrm\{end\}\}\}p\(t\)\\int\_\{\\Omega\_\{x\}\}q\(x\\mid t,\\mu\)\\cdot\\frac\{p\(x\)^\{2\}\}\{q\(x\\mid t,\\mu\)^\{2\}\}\\,r\_\{\\theta\}\(x,t;\\mu\)^\{4\}\\,dx\\,dt\\,d\\mu=𝔼t,μ​\[∫Ωxp​\(x\)2​rθ​\(x,t,μ\)4q⁡\(x∣t,μ\)​𝑑x\]\.\\displaystyle=\\mathbb\{E\}\_\{t,\\mu\}\\Bigg\[\\int\_\{\\Omega\_\{x\}\}\\frac\{p\(x\)^\{2\}\\,r\_\{\\theta\}\(x,t;\\mu\)^\{4\}\}\{q\(x\\mid t,\\mu\)\}\\,dx\\Bigg\]\.\(33\)Since∫Ωxq⁡\(x∣t,μ\)​𝑑x=1\\int\_\{\\Omega\_\{x\}\}q\(x\\mid t,\\mu\)\\,dx=1for any\(t,μ\)\(t,\\mu\), we multiply the integrand by this identity:

𝔼q​\[\(p⁡\(x\)q⁡\(x∣t,μ\)​rθ​\(x,t,μ\)2\)2\]\\displaystyle\\mathbb\{E\}\_\{q\}\\\!\\left\[\\left\(\\frac\{p\(x\)\}\{q\(x\\mid t,\\mu\)\}\\,r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\)^\{\\\!2\}\\,\\right\]=𝔼t,μ​\[\(∫Ωxp​\(x\)2​rθ​\(x,t,μ\)4q⁡\(x∣t,μ\)​𝑑x\)​\(∫Ωxq⁡\(x∣t,μ\)​𝑑x\)\]\\displaystyle=\\mathbb\{E\}\_\{t,\\mu\}\\Bigg\[\\left\(\\int\_\{\\Omega\_\{x\}\}\\frac\{p\(x\)^\{2\}\\,r\_\{\\theta\}\(x,t;\\mu\)^\{4\}\}\{q\(x\\mid t,\\mu\)\}\\,dx\\right\)\\left\(\\int\_\{\\Omega\_\{x\}\}q\(x\\mid t,\\mu\)\\,dx\\right\)\\Bigg\]=𝔼t,μ​\[\(∫Ωx\(p⁡\(x\)​rθ​\(x,t,μ\)2q⁡\(x∣t,μ\)\)2​𝑑x\)​\(∫Ωx\(q⁡\(x∣t,μ\)\)2​𝑑x\)\]\.\\displaystyle=\\mathbb\{E\}\_\{t,\\mu\}\\Bigg\[\\left\(\\int\_\{\\Omega\_\{x\}\}\\left\(\\frac\{p\(x\)\\,r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\}\{\\sqrt\{q\(x\\mid t,\\mu\)\}\}\\right\)^\{2\}\\,dx\\right\)\\left\(\\int\_\{\\Omega\_\{x\}\}\\left\(\\sqrt\{q\(x\\mid t,\\mu\)\}\\right\)^\{2\}\\,dx\\right\)\\Bigg\]\.\(34\)Applying the Cauchy\-Schwarz inequality to the inner spatial integral for each fixed\(t,μ\)\(t,\\mu\):

𝔼q​\[\(p⁡\(x\)q⁡\(x∣t,μ\)​rθ​\(x,t,μ\)2\)2\]\\displaystyle\\mathbb\{E\}\_\{q\}\\\!\\left\[\\left\(\\frac\{p\(x\)\}\{q\(x\\mid t,\\mu\)\}\\,r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\)^\{\\\!2\}\\,\\right\]≥𝔼t,μ​\[\(∫Ωxp⁡\(x\)​rθ​\(x,t,μ\)2q⁡\(x∣t,μ\)⋅q⁡\(x∣t,μ\)​𝑑x\)2\]\\displaystyle\\geq\\mathbb\{E\}\_\{t,\\mu\}\\Bigg\[\\left\(\\int\_\{\\Omega\_\{x\}\}\\frac\{p\(x\)\\,r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\}\{\\sqrt\{q\(x\\mid t,\\mu\)\}\}\\cdot\\sqrt\{q\(x\\mid t,\\mu\)\}\\,dx\\right\)^\{2\}\\Bigg\]=𝔼t,μ​\[\(∫Ωxp⁡\(x\)​rθ​\(x,t,μ\)2​𝑑x\)2\]\\displaystyle=\\mathbb\{E\}\_\{t,\\mu\}\\Bigg\[\\left\(\\int\_\{\\Omega\_\{x\}\}p\(x\)\\,r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\,dx\\right\)^\{2\}\\Bigg\]=𝔼t,μ​\[\(𝔼x​\[rθ​\(x,t,μ\)2\]\)2\]\.\\displaystyle=\\mathbb\{E\}\_\{t,\\mu\}\\Bigg\[\\Big\(\\mathbb\{E\}\_\{x\}\[r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\]\\Big\)^\{2\}\\Bigg\]\.\(35\)Equality in Cauchy\-Schwarz holds if and only ifq⁡\(x∣t,μ\)∝p⁡\(x\)​rθ​\(x,t,μ\)2q\(x\\mid t,\\mu\)\\propto p\(x\)r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\. Thus, taking the infimum over allq∈𝒬cq\\in\\mathcal\{Q\}\_\{c\}, we getVc⋆​\(θ\):=infq∈𝒬cσq2​\(θ\)V\_\{\\mathrm\{c\}\}^\{\\star\}\(\\theta\):=\\inf\_\{q\\in\\mathcal\{Q\}\_\{c\}\}\\sigma\_\{q\}^\{2\}\(\\theta\):

Vc⋆​\(θ\)\+ℒ​\(θ\)2\\displaystyle V\_\{\\mathrm\{c\}\}^\{\\star\}\(\\theta\)\+\\mathcal\{L\}\(\\theta\)^\{2\}=𝔼t,μ​\[\(𝔼x​\[rθ​\(x,t,μ\)2\]\)2\]\\displaystyle=\\mathbb\{E\}\_\{t,\\mu\}\\Bigg\[\\Big\(\\mathbb\{E\}\_\{x\}\[r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\]\\Big\)^\{2\}\\Bigg\]\(36\)

##### Case 2: Unconditional Spatial Proposal \(q∈𝒬ncq\\in\\mathcal\{Q\}\_\{\\mathrm\{nc\}\}\)

Forq∈𝒬ncq\\in\\mathcal\{Q\}\_\{\\mathrm\{nc\}\}, the spatial proposalq⁡\(x∣t\)q\(x\\mid t\)is restricted to be independent ofμ\\mu\. Starting from the second moment in \([33](https://arxiv.org/html/2610.09126#A1.E33)\), sinceq⁡\(x∣t\)q\(x\\mid t\)is independent ofμ\\mu, we can pass the expectation overμ\\muinside the spatial integral:

𝔼q​\[\(p⁡\(x\)q⁡\(x∣t\)​rθ​\(x,t,μ\)2\)2\]\\displaystyle\\mathbb\{E\}\_\{q\}\\\!\\left\[\\left\(\\frac\{p\(x\)\}\{q\(x\\mid t\)\}\\,r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\)^\{\\\!2\}\\,\\right\]=𝔼t​\[∫Ωxp​\(x\)2​𝔼μ​\[rθ​\(x,t,μ\)4\]q⁡\(x∣t\)​𝑑x\]\.\\displaystyle=\\mathbb\{E\}\_\{t\}\\Bigg\[\\int\_\{\\Omega\_\{x\}\}\\frac\{p\(x\)^\{2\}\\,\\mathbb\{E\}\_\{\\mu\}\[r\_\{\\theta\}\(x,t;\\mu\)^\{4\}\]\}\{q\(x\\mid t\)\}\\,dx\\Bigg\]\.\(37\)Using the identity∫Ωxq⁡\(x∣t\)​𝑑x=1\\int\_\{\\Omega\_\{x\}\}q\(x\\mid t\)\\,dx=1for eachtt:

𝔼q​\[\(p⁡\(x\)q⁡\(x∣t\)​rθ​\(x,t,μ\)2\)2\]\\displaystyle\\mathbb\{E\}\_\{q\}\\\!\\left\[\\left\(\\frac\{p\(x\)\}\{q\(x\\mid t\)\}\\,r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\)^\{\\\!2\}\\,\\right\]=𝔼t​\[\(∫Ωxp​\(x\)2​𝔼μ​\[rθ​\(x,t,μ\)4\]q⁡\(x∣t\)​𝑑x\)​\(∫Ωxq⁡\(x∣t\)​𝑑x\)\]\\displaystyle=\\mathbb\{E\}\_\{t\}\\Bigg\[\\left\(\\int\_\{\\Omega\_\{x\}\}\\frac\{p\(x\)^\{2\}\\,\\mathbb\{E\}\_\{\\mu\}\[r\_\{\\theta\}\(x,t;\\mu\)^\{4\}\]\}\{q\(x\\mid t\)\}\\,dx\\right\)\\left\(\\int\_\{\\Omega\_\{x\}\}q\(x\\mid t\)\\,dx\\right\)\\Bigg\]=𝔼t​\[\(∫Ωx\(p⁡\(x\)​𝔼μ​\[rθ​\(x,t,μ\)4\]q⁡\(x∣t\)\)2​𝑑x\)​\(∫Ωx\(q⁡\(x∣t\)\)2​𝑑x\)\]\.\\displaystyle=\\mathbb\{E\}\_\{t\}\\Bigg\[\\left\(\\int\_\{\\Omega\_\{x\}\}\\left\(\\frac\{p\(x\)\\sqrt\{\\mathbb\{E\}\_\{\\mu\}\[r\_\{\\theta\}\(x,t;\\mu\)^\{4\}\]\}\}\{\\sqrt\{q\(x\\mid t\)\}\}\\right\)^\{2\}dx\\right\)\\left\(\\int\_\{\\Omega\_\{x\}\}\\left\(\\sqrt\{q\(x\\mid t\)\}\\right\)^\{2\}dx\\right\)\\Bigg\]\.\(38\)Applying the Cauchy\-Schwarz inequality overΩx\\Omega\_\{x\}for eachtt:

𝔼q​\[\(p⁡\(x\)q⁡\(x∣t\)​rθ​\(x,t,μ\)2\)2\]\\displaystyle\\mathbb\{E\}\_\{q\}\\\!\\left\[\\left\(\\frac\{p\(x\)\}\{q\(x\\mid t\)\}\\,r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\)^\{\\\!2\}\\,\\right\]≥𝔼t​\[\(∫Ωxp⁡\(x\)​𝔼μ​\[rθ​\(x,t,μ\)4\]​𝑑x\)2\]\\displaystyle\\geq\\mathbb\{E\}\_\{t\}\\Bigg\[\\left\(\\int\_\{\\Omega\_\{x\}\}p\(x\)\\sqrt\{\\mathbb\{E\}\_\{\\mu\}\[r\_\{\\theta\}\(x,t;\\mu\)^\{4\}\]\}\\,dx\\right\)^\{2\}\\Bigg\]=𝔼t​\[\(𝔼x​\[𝔼μ​\[rθ​\(x,t,μ\)4\]\]\)2\]\.\\displaystyle=\\mathbb\{E\}\_\{t\}\\Bigg\[\\left\(\\mathbb\{E\}\_\{x\}\\Big\[\\sqrt\{\\mathbb\{E\}\_\{\\mu\}\[r\_\{\\theta\}\(x,t;\\mu\)^\{4\}\]\}\\Big\]\\right\)^\{2\}\\Bigg\]\.\(39\)Equality holds whenq⁡\(x∣t\)∝p⁡\(x\)​𝔼μ​\[rθ​\(x,t,μ\)4\]q\(x\\mid t\)\\propto p\(x\)\\sqrt\{\\mathbb\{E\}\_\{\\mu\}\[r\_\{\\theta\}\(x,t;\\mu\)^\{4\}\]\}\. Hence, taking the infimum over allq∈𝒬ncq\\in\\mathcal\{Q\}\_\{\\mathrm\{nc\}\}, we obtainVnc⋆​\(θ\):=infq∈𝒬ncσq2​\(θ\)V\_\{\\mathrm\{nc\}\}^\{\\star\}\(\\theta\):=\\inf\_\{q\\in\\mathcal\{Q\}\_\{\\mathrm\{nc\}\}\}\\sigma\_\{q\}^\{2\}\(\\theta\):

Vnc⋆​\(θ\)\+ℒ​\(θ\)2\\displaystyle V\_\{\\mathrm\{nc\}\}^\{\\star\}\(\\theta\)\+\\mathcal\{L\}\(\\theta\)^\{2\}=𝔼t​\[\(𝔼x​\[𝔼μ​\[rθ​\(x,t,μ\)4\]\]\)2\]\\displaystyle=\\mathbb\{E\}\_\{t\}\\Bigg\[\\left\(\\mathbb\{E\}\_\{x\}\\Big\[\\sqrt\{\\mathbb\{E\}\_\{\\mu\}\[r\_\{\\theta\}\(x,t;\\mu\)^\{4\}\]\}\\Big\]\\right\)^\{2\}\\Bigg\]\(40\)

##### Comparison via Minkowski’s Integral Inequality

To compareVc⋆​\(θ\)V\_\{\\mathrm\{c\}\}^\{\\star\}\(\\theta\)andVnc⋆​\(θ\)V\_\{\\mathrm\{nc\}\}^\{\\star\}\(\\theta\), we express the inner parameter expectations using theL2​\(μ\)L^\{2\}\(\\mu\)\-norm of the squared residualrθ​\(x,t,μ\)2r\_\{\\theta\}\(x,t;\\mu\)^\{2\}:

‖rθ​\(x,t,μ\)2‖L2​\(μ\):=𝔼μ​\[rθ​\(x,t,μ\)4\]\.\\Big\\\|r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\Big\\\|\_\{L^\{2\}\(\\mu\)\}:=\\sqrt\{\\mathbb\{E\}\_\{\\mu\}\\left\[r\_\{\\theta\}\(x,t;\\mu\)^\{4\}\\right\]\}\.\(41\)
Rewriting \([36](https://arxiv.org/html/2610.09126#A1.E36)\) and \([40](https://arxiv.org/html/2610.09126#A1.E40)\) side by side:

Vc⋆​\(θ\)\+ℒ​\(θ\)2=𝔼t​\[‖𝔼x​\[rθ​\(x,t,μ\)2\]‖L2​\(μ\)2\],Vnc⋆​\(θ\)\+ℒ​\(θ\)2=𝔼t​\[\(𝔼x​\[‖rθ​\(x,t,μ\)2‖L2​\(μ\)\]\)2\]\.V\_\{\\mathrm\{c\}\}^\{\\star\}\(\\theta\)\+\\mathcal\{L\}\(\\theta\)^\{2\}=\\mathbb\{E\}\_\{t\}\\Bigg\[\\;\\left\\\|\\,\\mathbb\{E\}\_\{x\}\\\!\\left\[r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\]\\right\\\|\_\{L^\{2\}\(\\mu\)\}^\{2\}\\;\\Bigg\],\\qquad V\_\{\\mathrm\{nc\}\}^\{\\star\}\(\\theta\)\+\\mathcal\{L\}\(\\theta\)^\{2\}=\\mathbb\{E\}\_\{t\}\\Bigg\[\\;\\left\(\\mathbb\{E\}\_\{x\}\\\!\\left\[\\,\\left\\\|r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\\\|\_\{L^\{2\}\(\\mu\)\}\\right\]\\right\)^\{2\}\\;\\Bigg\]\.\(42\)
The only difference betweenVc⋆V\_\{\\mathrm\{c\}\}^\{\\star\}andVnc⋆V\_\{\\mathrm\{nc\}\}^\{\\star\}is the order of the spatial expectation𝔼x\\mathbb\{E\}\_\{x\}and theL2​\(μ\)L^\{2\}\(\\mu\)\-norm: taking the spatial average first and then measuring the parameter norm, versus measuring the parameter norm first and then taking the spatial average\.

By Minkowski’s integral inequality:

‖𝔼x​\[rθ​\(x,t,μ\)2\]‖L2​\(μ\)≤𝔼x​\[‖rθ​\(x,t,μ\)2‖L2​\(μ\)\]\.\\left\\\|\\,\\mathbb\{E\}\_\{x\}\\\!\\left\[r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\]\\right\\\|\_\{L^\{2\}\(\\mu\)\}\\;\\leq\\;\\mathbb\{E\}\_\{x\}\\\!\\left\[\\,\\left\\\|r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\\\|\_\{L^\{2\}\(\\mu\)\}\\right\]\.\(43\)
Squaring both sides of \([43](https://arxiv.org/html/2610.09126#A1.E43)\) and taking the expectation w\.r\.t timettyields:

𝔼t​\[‖𝔼x​\[rθ​\(x,t,μ\)2\]‖L2​\(μ\)2\]≤𝔼t​\[\(𝔼x​\[‖rθ​\(x,t,μ\)2‖L2​\(μ\)\]\)2\]\.\\mathbb\{E\}\_\{t\}\\Bigg\[\\;\\left\\\|\\,\\mathbb\{E\}\_\{x\}\\\!\\left\[r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\]\\right\\\|\_\{L^\{2\}\(\\mu\)\}^\{2\}\\;\\Bigg\]\\;\\leq\\;\\mathbb\{E\}\_\{t\}\\Bigg\[\\;\\left\(\\mathbb\{E\}\_\{x\}\\\!\\left\[\\,\\left\\\|r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\right\\\|\_\{L^\{2\}\(\\mu\)\}\\right\]\\right\)^\{2\}\\;\\Bigg\]\.\(44\)
Subtractingℒ​\(θ\)2\\mathcal\{L\}\(\\theta\)^\{2\}from both sides gives the desired inequality:

Vc⋆​\(θ\)≤Vnc⋆​\(θ\)for every​θ∈Θ\.V\_\{\\mathrm\{c\}\}^\{\\star\}\(\\theta\)\\;\\leq\\;V\_\{\\mathrm\{nc\}\}^\{\\star\}\(\\theta\)\\qquad\\text\{for every \}\\theta\\in\\Theta\.\(45\)∎

### A\.6Proof of Remark[2\.1](https://arxiv.org/html/2610.09126#S2.Thmremark1)\.

We first recall the equality condition for the continuous Minkowski inequality \([43](https://arxiv.org/html/2610.09126#A1.E43)\)\. In general, equality in the vector triangle inequality‖f\+g‖=‖f‖\+‖g‖\\\|f\+g\\\|=\\\|f\\\|\+\\\|g\\\|holds if and only ifffandggare linearly dependent with non\-negative scalars \(i\.e\., they point in the same direction\)\.

Extending this to the continuous Minkowski inequality in theL2​\(μ\)L^\{2\}\(\\mu\)\-norm over the parameter space𝒫\\mathcal\{P\}, equality holds in \([43](https://arxiv.org/html/2610.09126#A1.E43)\) at a fixed timettif and only if all spatial functions in the family\{x↦rθ\(x,t;μ\)2\}μ∈𝒫\\\{x\\mapsto r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\\}\_\{\\mu\\in\\mathcal\{P\}\}are non\-negative scalar multiples of one single spatial functionat​\(x\)a\_\{t\}\(x\)\. That is, for almost everytt, there must exist a spatial factorat​\(x\)≥0a\_\{t\}\(x\)\\geq 0and aμ\\mu\-dependent factorct​\(μ\)≥0c\_\{t\}\(\\mu\)\\geq 0such that the squared residual factors into a product:

rθ​\(x,t,μ\)2=at​\(x\)​ct​\(μ\)\.r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\;=\\;a\_\{t\}\(x\)\\,c\_\{t\}\(\\mu\)\.\(46\)
To see how condition \([46](https://arxiv.org/html/2610.09126#A1.E46)\) relates to the spatial shape of the residual, consider its normalized spatial distribution:

ϱθ​\(x∣t,μ\):=p⁡\(x\)​rθ​\(x,t,μ\)2∫Ωxp⁡\(x′\)​rθ​\(x′,t,μ\)2​d​x′\.\\varrho\_\{\\theta\}\(x\\mid t,\\mu\)\\;:=\\;\\frac\{p\(x\)\\,r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\}\{\\int\_\{\\Omega\_\{x\}\}p\(x^\{\\prime\}\)\\,r\_\{\\theta\}\(x^\{\\prime\},t;\\mu\)^\{2\}\\,dx^\{\\prime\}\}\.\(47\)Substituting the factorized form \([46](https://arxiv.org/html/2610.09126#A1.E46)\) into both the numerator and the denominator of \([47](https://arxiv.org/html/2610.09126#A1.E47)\) yields:

ϱθ​\(x∣t,μ\)\\displaystyle\\varrho\_\{\\theta\}\(x\\mid t,\\mu\)=p⁡\(x\)​\(at​\(x\)​ct​\(μ\)\)∫Ωxp⁡\(x′\)​\(at​\(x′\)​ct​\(μ\)\)​d​x′\\displaystyle\\;=\\;\\frac\{p\(x\)\\,\\big\(a\_\{t\}\(x\)\\,c\_\{t\}\(\\mu\)\\big\)\}\{\\int\_\{\\Omega\_\{x\}\}p\(x^\{\\prime\}\)\\,\\big\(a\_\{t\}\(x^\{\\prime\}\)\\,c\_\{t\}\(\\mu\)\\big\)\\,dx^\{\\prime\}\}\(48\)=ct​\(μ\)⋅\(p⁡\(x\)​at​\(x\)\)ct​\(μ\)⋅∫Ωxp⁡\(x′\)​at​\(x′\)​d​x′\\displaystyle\\;=\\;\\frac\{c\_\{t\}\(\\mu\)\\cdot\\big\(p\(x\)\\,a\_\{t\}\(x\)\\big\)\}\{c\_\{t\}\(\\mu\)\\cdot\\int\_\{\\Omega\_\{x\}\}p\(x^\{\\prime\}\)\\,a\_\{t\}\(x^\{\\prime\}\)\\,dx^\{\\prime\}\}\(49\)=p⁡\(x\)​at​\(x\)∫Ωxp⁡\(x′\)​at​\(x′\)​d​x′\.\\displaystyle\\;=\\;\\frac\{p\(x\)\\,a\_\{t\}\(x\)\}\{\\int\_\{\\Omega\_\{x\}\}p\(x^\{\\prime\}\)\\,a\_\{t\}\(x^\{\\prime\}\)\\,dx^\{\\prime\}\}\.\(50\)Since theμ\\mu\-dependent scalarct​\(μ\)c\_\{t\}\(\\mu\)cancels out entirely, the resulting density depends solely onxxandtt, making it completely independent ofμ\\mu\.

Conversely, ifϱθ​\(x∣t,μ\)\\varrho\_\{\\theta\}\(x\\mid t,\\mu\)is independent ofμ\\mu, we can writeϱθ​\(x∣t,μ\)=ϱθ​\(x∣t\)\\varrho\_\{\\theta\}\(x\\mid t,\\mu\)=\\varrho\_\{\\theta\}\(x\\mid t\)\. Rearranging definition \([47](https://arxiv.org/html/2610.09126#A1.E47)\) then gives:

rθ​\(x,t,μ\)2=ϱθ​\(x∣t\)p⁡\(x\)⋅\(∫Ωxp⁡\(x′\)​rθ​\(x′,t,μ\)2​d​x′\),r\_\{\\theta\}\(x,t;\\mu\)^\{2\}\\;=\\;\\frac\{\\varrho\_\{\\theta\}\(x\\mid t\)\}\{p\(x\)\}\\cdot\\left\(\\int\_\{\\Omega\_\{x\}\}p\(x^\{\\prime\}\)\\,r\_\{\\theta\}\(x^\{\\prime\},t;\\mu\)^\{2\}\\,dx^\{\\prime\}\\right\),\(51\)which is precisely in the separable format​\(x\)​ct​\(μ\)a\_\{t\}\(x\)\\,c\_\{t\}\(\\mu\)withat​\(x\)=ϱθ​\(x∣t\)/p⁡\(x\)a\_\{t\}\(x\)=\\varrho\_\{\\theta\}\(x\\mid t\)/p\(x\)andct​\(μ\)=∫Ωxp⁡\(x′\)​rθ​\(x′,t,μ\)2​d​x′c\_\{t\}\(\\mu\)=\\int\_\{\\Omega\_\{x\}\}p\(x^\{\\prime\}\)\\,r\_\{\\theta\}\(x^\{\\prime\},t;\\mu\)^\{2\}\\,dx^\{\\prime\}\.

Thus, equality holds in Theorem[2\.1](https://arxiv.org/html/2610.09126#S2.Thmtheorem1)if and only if the normalized spatial shapeϱθ​\(x∣t,μ\)\\varrho\_\{\\theta\}\(x\\mid t,\\mu\)carries no actualμ\\mu\-dependence\.

This mathematical equivalence highlights a critical physical insight: it is*not*sufficient for the residual magnitude to vary withμ\\mu\. Any overall magnitude variation is entirely absorbed byct​\(μ\)c\_\{t\}\(\\mu\), which an optimally tunedμ\\mu\-independent proposal can handle without performance degradation\. For strict separation to fail \(and equality to hold\), the residual must maintain a fixed spatial shape across allμ\\mu\.

###### Assumption A\.4\(Non\-degeneracy\)\.

For everyθ∈Θ\\theta\\in\\Theta, the normalized spatial shapeϱθ\(⋅∣t,μ\)\\varrho\_\{\\theta\}\(\\cdot\\mid t,\\mu\)is notp⁡\(μ\)p\(\\mu\)\-a\.e\. constant inμ\\muon a set of timesttwith positivep⁡\(t\)p\(t\)\-measure\.

Under Assumption[A\.4](https://arxiv.org/html/2610.09126#A1.Thmassumption4), the condition for equality \([46](https://arxiv.org/html/2610.09126#A1.E46)\) is violated on a set of times of positive measure\. Consequently, the Minkowski inequality \([43](https://arxiv.org/html/2610.09126#A1.E43)\) becomes strict, and taking the expectation overttpreserves this strict inequality:

Vc⋆​\(θ\)<Vnc⋆​\(θ\)for every​θ∈Θ\.V\_\{\\mathrm\{c\}\}^\{\\star\}\(\\theta\)\\;<\\;V\_\{\\mathrm\{nc\}\}^\{\\star\}\(\\theta\)\\qquad\\text\{for every \}\\theta\\in\\Theta\.\(52\)
In metal AM thermal modeling, Assumption[A\.4](https://arxiv.org/html/2610.09126#A1.Thmassumption4)naturally holds due to the underlying physical dynamics of the system\. Variations in process conditionsμ\\mudirectly change the governing PDEs and BCs\. As a result, the entire*residual landscape*changes its spatial structure rather than just scaling up or down: localized high\-residual regions shift or reshape across the domain asμ\\muvaries\. Since a spatial profile cannot be expressed as a simple scalar multiple of a shifted or reshaped profile, the normalized shape \([47](https://arxiv.org/html/2610.09126#A1.E47)\) inherently varies withμ\\mu\. This structural non\-separability holds true regardless of the neural network’s architecture or approximation quality, as theμ\\mu\-dependent spatial variation is directly imprinted on the residual landscape by the physical problem itself\. This completes the proof of Remark[2\.1](https://arxiv.org/html/2610.09126#S2.Thmremark1)\. ∎

### A\.7Qualitative temperature fields comparison

![Refer to caption](https://arxiv.org/html/2610.09126v1/Fig11_fem_vs_proposed_fields_5mats_turbo_t0p5.png)Figure 11:Temperature field att=0\.5t=0\.5s for the five validation materials \(rows\): FEM reference and the proposed method, on the top surface and on they=5y=5mm cross\-section\. Note that each material uses its own color scale due to differences in thermophysical properties\.![Refer to caption](https://arxiv.org/html/2610.09126v1/Fig12_fem_vs_proposed_fields_5mats_turbo_t1p5.png)Figure 12:Temperature field att=1\.5t=1\.5s \(same layout and color scales as Figure[11](https://arxiv.org/html/2610.09126#A1.F11)\)\. The laser has advanced tox=20x=20mm and the trailing heat tail has developed\.![Refer to caption](https://arxiv.org/html/2610.09126v1/Fig13_fem_vs_proposed_fields_5mats_turbo_t2p5.png)Figure 13:Temperature field att=2\.5t=2\.5s \(same layout and color scales as Figure[11](https://arxiv.org/html/2610.09126#A1.F11)\)\. The proposed method reproduces the melt pool, the trailing heat tail, and the depth of heat penetration for every material, including the out\-of\-distribution AlSi10Mg and Copper\.
### A\.8Temperature fields deviation from FEM

![Refer to caption](https://arxiv.org/html/2610.09126v1/Fig14_deviation_fields_3mats_signed_Ti6Al4V_mix_dense_bc.png)Figure 14:Signed deviation and collocation overlay for Ti\-6Al\-4V \(kc=6\.7k\_\{c\}=6\.7W m\-1K\-1\), plotted as in Fig\.[7](https://arxiv.org/html/2610.09126#S4.F7)\.![Refer to caption](https://arxiv.org/html/2610.09126v1/Fig15_deviation_fields_3mats_signed_SS316L_mix_dense_bc.png)Figure 15:Signed deviation and collocation overlay for SS316L \(kc=16k\_\{c\}=16W m\-1K\-1\), plotted as in Fig\.[7](https://arxiv.org/html/2610.09126#S4.F7)\.![Refer to caption](https://arxiv.org/html/2610.09126v1/Fig16_deviation_fields_3mats_signed_AlSi10Mg_mix_dense_bc.png)Figure 16:Signed deviation and collocation overlay for AlSi10Mg \(kc=150k\_\{c\}=150W m\-1K\-1\), plotted as in Fig\.[7](https://arxiv.org/html/2610.09126#S4.F7)\.

## Acknowledgements

This work was supported by the National Research Foundation of Korea \(NRF\) grant funded by the Korea government \(MSIT\) \(RS\-2025\-02216260, RS\-2024\-00431837\)\.

## Data availability statement

## Generative Artificial Intelligence \(AI\)

The authors used Claude Sonnet 5 by Anthropic for grammar checking, polishing, and improving readability\. After using this tool, the authors reviewed and edited the content as needed, and acknowledge that the authors take full responsibility for the content of the publication\.

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