Physics-Informed Conformal Prediction: Embedding PDE Consistency into Distribution-Free Uncertainty Quantification for Neural Operators

# Embedding PDE Consistency into Distribution-Free UQ for Neural Operators
Source: [https://arxiv.org/html/2609.11935](https://arxiv.org/html/2609.11935)
## Physics\-Informed Conformal Prediction: Embedding PDE Consistency into Distribution\-Free UQ for Neural Operators

\(June 2026\)

###### Abstract

Neural operators such as the Fourier Neural Operator \(FNO\) achieve remarkable accuracy in approximating solutions to partial differential equations \(PDEs\)\. However, providing rigorous uncertainty estimates remains an open challenge\. We proposePhysics\-Informed Conformal Prediction \(PI\-CP\), a framework that embeds PDE residuals into the nonconformity score of split conformal prediction, producing prediction intervals that are \(i\) distribution\-free with provable coverage guarantees, and \(ii\) spatially adaptive when the PDE residual correlates with prediction error—tighter where physics is well\-satisfied, wider where it is violated\. Additionally, we prove that FNO’s translation equivariance creates a fundamental approximation barrier for PDEs with Dirichlet boundary conditions, and show that coordinate channels resolve this with up to63×63\\timeserror reduction\. We validate PI\-CP across six physics scenarios—heat conduction \(2D/3D\), structural mechanics \(2D/3D\), Darcy flow, and Navier\-Stokes—demonstrating consistent 89–91% coverage for all four Conformal methods, while MC Dropout and Deep Ensembles are unstable \(82–100%\)\. FNO outperforms CNN and DeepONet by1010–12×12\\times\.

## 1Introduction

Neural operators\(Li et al\.,[2021](https://arxiv.org/html/2609.11935#bib.bib1); Lu et al\.,[2021](https://arxiv.org/html/2609.11935#bib.bib2); Kovachki et al\.,[2023](https://arxiv.org/html/2609.11935#bib.bib13)\)learn mappings between infinite\-dimensional function spaces, offering orders\-of\-magnitude speedup over traditional PDE solvers after training\. The Fourier Neural Operator \(FNO\), in particular, achieves sub\-1% relative error on benchmark PDEs such as Darcy flow, Navier\-Stokes equations, and elasticity\. However, deploying these surrogates in safety\-critical engineering applications—bridge design, aircraft certification, nuclear reactor analysis—requires rigorous uncertainty quantification \(UQ\)\(Psaros et al\.,[2023](https://arxiv.org/html/2609.11935#bib.bib16)\): a point prediction without a calibrated confidence interval is of limited value when lives are at stake\.

#### The UQ challenge\.

Standard deep learning uncertainty methods \(MC Dropout\(Gal & Ghahramani,[2016](https://arxiv.org/html/2609.11935#bib.bib6)\), Deep Ensembles\(Lakshminarayanan et al\.,[2017](https://arxiv.org/html/2609.11935#bib.bib7)\)\) lack theoretical coverage guarantees\. As we demonstrate empirically \(§[5](https://arxiv.org/html/2609.11935#S5)\), MC Dropout coverage ranges from 82% to 100% depending on the scenario, providing no reliability contract\. Conformal prediction\(Vovk et al\.,[2005](https://arxiv.org/html/2609.11935#bib.bib3)\)solves this by providing distribution\-free, finite\-sample coverage guarantees under exchangeability\. However, standard split conformal prediction produces*spatially uniform*intervals—the same width everywhere—ignoring the physical structure of the problem\.

#### Key insight\.

In scientific computing, we have access to the governing PDE𝒟​\[u\]=0\\mathcal\{D\}\[u\]=0\. The PDE residualR​\(x\)=𝒟​\[f​\(x\)\]R\(x\)=\\mathcal\{D\}\[f\(x\)\]at a predictionf​\(x\)f\(x\)is a*physics\-aware error indicator*: regions whereR​\(x\)R\(x\)is large are more likely to contain prediction errors\. We embed this insight into conformal prediction by defining a nonconformity score normalized by the PDE residual, producing intervals that are \(i\) provably calibrated and \(ii\) spatially adaptive—tighter where the PDE is well\-satisfied, wider where physics is violated\.

#### A second insight: coordinate channels\.

While developing PI\-CP, we discovered that FNO’s translation equivariance—its core inductive bias—creates a fundamental approximation barrier for PDEs with Dirichlet boundary conditions\. On Darcy flow, FNO without coordinate channels achieves 63% error; adding grid coordinates\[x,y\]\[x,y\]reduces it to 1%, a 63×\\timesimprovement\. We provide a theoretical explanation based on symmetry analysis and validate across five scenarios\.

#### Contributions\.

1. 1\.Physics\-Informed Conformal Prediction \(PI\-CP\): A novel nonconformity scores=\|y−f​\(x\)\|/\(1\+λ​\|R​\(x\)\|\)s=\\left\|y\-f\(x\)\\right\|/\(1\+\\lambda\\left\|R\(x\)\\right\|\)incorporating PDE residualsR​\(x\)R\(x\), yielding intervals with provable coverage \(Proposition[1](https://arxiv.org/html/2609.11935#Thmproposition1)\) and spatial adaptivity conditioned on positive residual\-error correlation \(Remark[Remark](https://arxiv.org/html/2609.11935#Thmremarkx1)\)\.
2. 2\.Coordinate\-Aware FNO Theory: We prove that FNO’s translation equivariance creates an approximation barrier for Dirichlet BCs \(Theorem[1](https://arxiv.org/html/2609.11935#Thmtheorem1)\), and that coordinate channels break this symmetry to resolve the barrier \(Theorem[2](https://arxiv.org/html/2609.11935#Thmtheorem2)\), validated by up to63×63\\timesimprovement across five scenarios\.
3. 3\.Multi\-Physics Validation: PI\-CP across six physics scenarios with consistent 89–91% coverage for all Conformal methods\. FNO outperforms CNN and DeepONet by1010–12×12\\times\.

## 2Related Work

#### Neural Operators\.

The Fourier Neural Operator \(FNO\)\(Li et al\.,[2021](https://arxiv.org/html/2609.11935#bib.bib1)\)learns a mapping between function spaces via spectral convolutions in the frequency domain, achieving discretization\-invariant approximations\. DeepONet\(Lu et al\.,[2021](https://arxiv.org/html/2609.11935#bib.bib2)\)uses a branch\-trunk decomposition, where the branch network encodes the input function and the trunk network encodes the query locations\. Graph Neural Operators\(Li et al\.,[2020](https://arxiv.org/html/2609.11935#bib.bib11)\)extend this to irregular meshes\. While these methods achieve remarkable accuracy, they produce point predictions without uncertainty estimates\. Our work adds rigorous UQ*post hoc*, without modifying the training procedure, making it compatible with any pre\-trained neural operator\.

#### Conformal Prediction\.

Split conformal prediction\(Vovk et al\.,[2005](https://arxiv.org/html/2609.11935#bib.bib3); Shafer & Vovk,[2008](https://arxiv.org/html/2609.11935#bib.bib15)\)provides finite\-sample, distribution\-free marginal coverage under exchangeability\.Lei et al\. \([2018](https://arxiv.org/html/2609.11935#bib.bib4)\)introduced locally adaptive CP using normalized nonconformity scoress=\|y−f​\(x\)\|/σ​\(x\)s=\\left\|y\-f\(x\)\\right\|/\\sigma\(x\), whereσ​\(x\)\\sigma\(x\)estimates local prediction difficulty \(e\.g\., MC Dropout variance\)\. Conformalized Quantile Regression \(CQR\)\(Romano et al\.,[2019](https://arxiv.org/html/2609.11935#bib.bib8)\)achieves adaptivity via quantile regression\. Bates et al\.\(Bates et al\.,[2021](https://arxiv.org/html/2609.11935#bib.bib18)\)extend conformal methods to control general statistical risks beyond coverage\. For a comprehensive treatment, seeAngelopoulos & Bates \([2023](https://arxiv.org/html/2609.11935#bib.bib14)\)\.

While PI\-CP shares the mathematical structure of normalized scores\(Lei et al\.,[2018](https://arxiv.org/html/2609.11935#bib.bib4)\), the*information source*is fundamentally different\. Locally adaptive CP relies on model\-internal uncertaintyσ​\(x\)\\sigma\(x\); PI\-CP uses the PDE residual\|R​\(x\)\|\\left\|R\(x\)\\right\|—an*external*physics\-based signal computed from the known governing equations\. This distinction is consequential for neural operators: well\-trained FNO models produce near\-uniform dropout variance \(σ​\(x\)≈const\\sigma\(x\)\\approx\\text\{const\}\), yielding zero spatial adaptivity for normalized CP \(CV==0\.000, Table[2](https://arxiv.org/html/2609.11935#S5.T2)\)\. PDE residuals, by contrast, capture the spatially\-varying structure of physical inconsistency, providing a meaningful adaptivity signal \(CV==0\.221\)\. Furthermore,\|R​\(x\)\|\\left\|R\(x\)\\right\|requires no auxiliary uncertainty model—it is available for free in any PDE\-governed problem\.

#### Physics\-Informed ML\.

Physics\-Informed Neural Networks \(PINNs\)\(Raissi et al\.,[2019](https://arxiv.org/html/2609.11935#bib.bib5)\)embed PDE constraints into the*training*loss, guiding the model toward physically consistent solutions\. Our approach is complementary: we use PDE residuals at*inference*time to calibrate uncertainty, without retraining\. The two approaches can be combined—a PINN\-trained model could use PI\-CP for calibrated UQ\.

#### Equivariance in Neural Networks\.

Equivariant neural networks\(Cohen & Welling,[2016](https://arxiv.org/html/2609.11935#bib.bib9); Bronstein et al\.,[2017](https://arxiv.org/html/2609.11935#bib.bib10)\)exploit symmetry structures to improve generalization\. FNO’s translation equivariance is a key inductive bias for PDE problems on periodic domains\. However,Li et al\. \([2021](https://arxiv.org/html/2609.11935#bib.bib1)\)noted that “grid features” can be optionally added without analysis\. We show that for Dirichlet BCs, coordinate channels are not optional but*essential*, providing a group\-theoretic explanation and63×63\\timesempirical validation\.

## 3Background

### 3\.1Fourier Neural Operator

The FNO architecture applies spectral convolutions:

vt\+1​\(x\)=σ​\(W​vt​\(x\)\+ℱ−1​\[Rκ⋅v^t​\(κ\)\]​\(x\)\)v\_\{t\+1\}\(x\)=\\sigma\\\!\\left\(Wv\_\{t\}\(x\)\+\\mathcal\{F\}^\{\-1\}\[R\_\{\\kappa\}\\cdot\\hat\{v\}\_\{t\}\(\\kappa\)\]\(x\)\\right\)\(1\)whereRκR\_\{\\kappa\}is a learnable weight tensor applied to Fourier coefficientsv^t​\(κ\)\\hat\{v\}\_\{t\}\(\\kappa\), truncated to the lowestmmmodes\.

### 3\.2Split Conformal Prediction

Given calibration set\{\(xi,yi\)\}i=1n\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{n\}and significance levelα\\alpha:

1. 1\.Compute nonconformity scoressi=\|yi−f​\(xi\)\|s\_\{i\}=\\left\|y\_\{i\}\-f\(x\_\{i\}\)\\right\|
2. 2\.Find quantileq^=\\hat\{q\}=⌈\(1−α\)​\(n\+1\)⌉\\lceil\(1\-\\alpha\)\(n\+1\)\\rceil\-th order statistic
3. 3\.Output intervalC​\(x\)=\[f​\(x\)−q^,f​\(x\)\+q^\]C\(x\)=\[f\(x\)\-\\hat\{q\},\\;f\(x\)\+\\hat\{q\}\]

### 3\.3Notation

We writex∈Ω⊂ℝdx\\in\\Omega\\subset\\mathbb\{R\}^\{d\}for the spatial coordinate on domainΩ\\Omegawith boundary∂Ω\\partial\\Omega\. A PDE is written as𝒟​\[u\]​\(x\)=0\\mathcal\{D\}\[u\]\(x\)=0with differential operator𝒟\\mathcal\{D\}, subject to boundary conditionsu\|∂Ω=gu\|\_\{\\partial\\Omega\}=g\. The coefficient\-to\-solution operator isS:k↦uS:k\\mapsto u\. A neural operator approximation isfθf\_\{\\theta\}\(we dropθ\\thetawhen clear\)\. The PDE residual at a prediction isR​\(x\)=𝒟​\[f\]​\(x\)R\(x\)=\\mathcal\{D\}\[f\]\(x\)\. For conformal prediction, the significance level isα∈\(0,1\)\\alpha\\in\(0,1\), the nonconformity score isss, and the calibration quantile isq^\\hat\{q\}\. We use∥⋅∥L2\\left\\\|\\cdot\\right\\\|\_\{L^\{2\}\}for theL2​\(Ω\)L^\{2\}\(\\Omega\)norm\. Translation byδ\\deltais\(τδ​f\)​\(x\):=f​\(x−δ\)\(\\tau\_\{\\delta\}f\)\(x\):=f\(x\-\\delta\)\. The Fourier transform and its inverse areℱ\\mathcal\{F\}andℱ−1\\mathcal\{F\}^\{\-1\}\.

### 3\.4Problem Setup

We consider the operator learning setting: given a PDE𝒟​\[u\]=0\\mathcal\{D\}\[u\]=0with boundary conditions, the solution operatorSSmaps coefficient fieldsk∈𝒳k\\in\\mathcal\{X\}to solutionsu∈𝒴u\\in\\mathcal\{Y\}\. A neural operatorfθf\_\{\\theta\}approximatesSS\. We address two problems:

1. 1\.Uncertainty quantification: For a new inputxtestx\_\{\\text\{test\}\}, construct a prediction intervalC​\(xtest\)C\(x\_\{\\text\{test\}\}\)such thatP​\(ytest∈C​\(xtest\)\)≥1−αP\(y\_\{\\text\{test\}\}\\in C\(x\_\{\\text\{test\}\}\)\)\\geq 1\-\\alpha, whereytest=S​\(xtest\)y\_\{\\text\{test\}\}=S\(x\_\{\\text\{test\}\}\)\. The interval should be*spatially adaptive*—reflecting local prediction difficulty\.
2. 2\.Architectural limitation: For PDEs with non\-trivial Dirichlet BCs,SSis not translation\-equivariant\. How does FNO’s equivariance affect its ability to approximateSS, and how can this be resolved?

## 4Method

### 4\.1Physics\-Informed Nonconformity Score

#### Motivation\.

Standard split CP assigns the same interval width to all test points, regardless of their physical characteristics\. In scientific computing, however, we have a powerful*a priori*error indicator: the PDE residual\. If a predictionf​\(x\)f\(x\)satisfies the governing equations \(low\|R​\(x\)\|\|R\(x\)\|\), it is more likely to be accurate; if it violates physics \(high\|R​\(x\)\|\|R\(x\)\|\), it likely contains errors\. PI\-CP exploits this by normalizing the nonconformity score by\|R​\(x\)\|\|R\(x\)\|\.

###### Definition 1\(PDE Residual\)\.

For a PDE𝒟​\[u\]​\(x\)=0\\mathcal\{D\}\[u\]\(x\)=0with differential operator𝒟\\mathcal\{D\}, the residual at predictionf​\(x\)f\(x\)isR​\(x\)=𝒟​\[f\]​\(x\)R\(x\)=\\mathcal\{D\}\[f\]\(x\)\.

###### Definition 2\(PI\-CP Score\)\.

The physics\-informed nonconformity score is:

s​\(x,y\)=\|y−f​\(x\)\|1\+λ​\|R​\(x\)\|s\(x,y\)=\\frac\{\\left\|y\-f\(x\)\\right\|\}\{1\+\\lambda\\left\|R\(x\)\\right\|\}\(2\)whereλ≥0\\lambda\\geq 0controls physics influence\.

#### Interpretation\.

Where\|R​\(x\)\|\|R\(x\)\|is large \(physics violated\), the denominator inflates, reducing the scoress—effectively telling CP that large errors are “expected” here and should not widen the global quantile\. Conversely, where\|R​\(x\)\|≈0\|R\(x\)\|\\approx 0\(physics satisfied\), the score approaches the raw error\|y−f​\(x\)\|\|y\-f\(x\)\|, making the quantile tight\. The resulting interval is:

C​\(x\)=\[f​\(x\)−q^​\(1\+λ​\|R​\(x\)\|\),f​\(x\)\+q^​\(1\+λ​\|R​\(x\)\|\)\]C\(x\)=\\big\[f\(x\)\-\\hat\{q\}\(1\+\\lambda\\left\|R\(x\)\\right\|\),\\;\\;f\(x\)\+\\hat\{q\}\(1\+\\lambda\\left\|R\(x\)\\right\|\)\\big\]\(3\)This interval isspatially adaptive: wider where physics is violated, tighter where it is satisfied\. Theλ\\lambdaparameter smoothly interpolates between standard CP \(λ=0\\lambda=0\) and fully physics\-adaptive intervals \(λ→∞\\lambda\\to\\infty\)\.

###### Proposition 1\(Coverage Guarantee\)\.

Under exchangeability of\(xi,yi\)\(x\_\{i\},y\_\{i\}\)and a fixed trained modelff, PI\-CP satisfiesP​\(y∈C​\(x\)\)≥1−α\.P\\\!\\big\(y\\in C\(x\)\\big\)\\geq 1\-\\alpha\.

###### Proof\.

The scores​\(x,y\)=\|y−f​\(x\)\|/\(1\+λ​\|R​\(x\)\|\)s\(x,y\)=\\left\|y\-f\(x\)\\right\|/\(1\+\\lambda\\left\|R\(x\)\\right\|\)is a deterministic function of\(x,y\)\(x,y\)sinceff,λ\\lambda, and the differential operator𝒟\\mathcal\{D\}are all fixed at inference time\. Therefore exchangeability of\{\(xi,yi\)\}\\\{\(x\_\{i\},y\_\{i\}\)\\\}implies exchangeability of\{si\}\\\{s\_\{i\}\\\}, and the standard split CP argument\(Vovk et al\.,[2005](https://arxiv.org/html/2609.11935#bib.bib3)\)yields the coverage guarantee\. ∎

###### Proposition 2\(Interval Width Analysis\)\.

Letwstd​\(x\)=2​q^stdw\_\{\\text\{std\}\}\(x\)=2\\hat\{q\}\_\{\\text\{std\}\}be the interval width of standard CP \(constant acrossxx\), andwPI​\(x\)=2​q^PI​\(1\+λ​\|R​\(x\)\|\)w\_\{\\text\{PI\}\}\(x\)=2\\hat\{q\}\_\{\\text\{PI\}\}\(1\+\\lambda\\left\|R\(x\)\\right\|\)be the PI\-CP width\. Then:

1. 1\.Spatial adaptivity:wPI​\(x\)w\_\{\\text\{PI\}\}\(x\)varies with\|R​\(x\)\|\\left\|R\(x\)\\right\|, achieving width coefficient of variationCV=std​\(\|R\|\)/mean​\(\|R\|\)\\text\{CV\}=\\text\{std\}\(\\left\|R\\right\|\)/\\text\{mean\}\(\\left\|R\\right\|\), which is independent ofq^\\hat\{q\}\.
2. 2\.Bounded overhead: The expected width ratio satisfies𝔼​\[wPI\]/𝔼​\[wstd\]=\(q^PI/q^std\)​\(1\+λ​𝔼​\[\|R\|\]\)≤1\+λ​𝔼​\[\|R\|\]\\mathbb\{E\}\[w\_\{\\text\{PI\}\}\]/\\mathbb\{E\}\[w\_\{\\text\{std\}\}\]=\(\\hat\{q\}\_\{\\text\{PI\}\}/\\hat\{q\}\_\{\\text\{std\}\}\)\\big\(1\+\\lambda\\,\\mathbb\{E\}\[\\left\|R\\right\|\]\\big\)\\leq 1\+\\lambda\\,\\mathbb\{E\}\[\\left\|R\\right\|\], where the inequality uses Part \(3\)\.
3. 3\.Score calibration: PI\-CP’s quantileq^PI≤q^std\\hat\{q\}\_\{\\text\{PI\}\}\\leq\\hat\{q\}\_\{\\text\{std\}\}because normalizing by\(1\+λ​\|R\|\)≥1\(1\+\\lambda\\left\|R\\right\|\)\\geq 1compresses the score distribution, so the calibration quantile decreases\.

###### Proof\.

\(1\) follows directly fromwPI∝\(1\+λ​\|R\|\)w\_\{\\text\{PI\}\}\\propto\(1\+\\lambda\\left\|R\\right\|\)\. \(2\) Since𝔼​\[wPI\]=2​q^PI​\(1\+λ​𝔼​\[\|R\|\]\)\\mathbb\{E\}\[w\_\{\\text\{PI\}\}\]=2\\hat\{q\}\_\{\\text\{PI\}\}\(1\+\\lambda\\mathbb\{E\}\[\\left\|R\\right\|\]\)and𝔼​\[wstd\]=2​q^std\\mathbb\{E\}\[w\_\{\\text\{std\}\}\]=2\\hat\{q\}\_\{\\text\{std\}\}, the ratio equals\(q^PI/q^std\)​\(1\+λ​𝔼​\[\|R\|\]\)\(\\hat\{q\}\_\{\\text\{PI\}\}/\\hat\{q\}\_\{\\text\{std\}\}\)\(1\+\\lambda\\mathbb\{E\}\[\\left\|R\\right\|\]\), which is≤1\+λ​𝔼​\[\|R\|\]\\leq 1\+\\lambda\\mathbb\{E\}\[\\left\|R\\right\|\]by Part \(3\)\. \(3\) SincesPI=\|y−f\|/\(1\+λ​\|R\|\)≤\|y−f\|=sstds\_\{\\text\{PI\}\}=\\left\|y\-f\\right\|/\(1\+\\lambda\\left\|R\\right\|\)\\leq\\left\|y\-f\\right\|=s\_\{\\text\{std\}\}pointwise, the order statistics satisfyq^PI≤q^std\\hat\{q\}\_\{\\text\{PI\}\}\\leq\\hat\{q\}\_\{\\text\{std\}\}\. ∎

#### Interpretation\.

Proposition[2](https://arxiv.org/html/2609.11935#Thmproposition2)reveals that PI\-CP’s spatial adaptivity comes from two complementary effects: \(i\) the*local*widening factor\(1\+λ​\|R​\(x\)\|\)\(1\+\\lambda\\left\|R\(x\)\\right\|\)creates heterogeneous widths, and \(ii\) the*global*quantileq^PI\\hat\{q\}\_\{\\text\{PI\}\}shrinks relative to standard CP because the score distribution is compressed\. The net effect is that average width inflation is bounded \(typically<<5% as shown in Table[2](https://arxiv.org/html/2609.11935#S5.T2)\), while spatial variation can be substantial \(CV up to 0\.64 atλ=5\\lambda\{=\}5\)\. This is the key theoretical advantage of PI\-CP over standard CP: physics\-informed scoring redistributes interval width without proportional inflation\.

### 4\.2PI\-Normalized CP

Combining with epistemic uncertaintyσ​\(x\)\\sigma\(x\)from MC Dropout:

sPI\-N​\(x,y\)=\|y−f​\(x\)\|\(1\+λ​\|R​\(x\)\|\)⋅σ​\(x\)s\_\{\\text\{PI\-N\}\}\(x,y\)=\\frac\{\\left\|y\-f\(x\)\\right\|\}\{\(1\+\\lambda\\left\|R\(x\)\\right\|\)\\cdot\\sigma\(x\)\}\(4\)
Algorithm 1Physics\-Informed Conformal Prediction0:Trained model

ff, calibration set

\{\(xi,yi\)\}i=1n\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{n\}, test point

xtestx\_\{\\text\{test\}\},

α\\alpha,

λ\\lambda
1:Phase 1: Compute PDE residuals

2:for

i=1i=1to

nndo

3:

Ri←𝒟​\[f​\(xi\)\]R\_\{i\}\\leftarrow\\mathcal\{D\}\[f\(x\_\{i\}\)\]// Apply PDE operator to prediction

4:endfor

5:Phase 2: Compute nonconformity scores

6:for

i=1i=1to

nndo

7:

si←\|yi−f​\(xi\)\|/\(1\+λ​\|Ri\|\)s\_\{i\}\\leftarrow\\left\|y\_\{i\}\-f\(x\_\{i\}\)\\right\|/\(1\+\\lambda\\left\|R\_\{i\}\\right\|\)
8:endfor

9:Phase 3: Find quantile

10:

q^←\\hat\{q\}\\leftarrow⌈\(1−α\)​\(n\+1\)⌉\\lceil\(1\-\\alpha\)\(n\+1\)\\rceil\-th order statistic of

\{si\}\\\{s\_\{i\}\\\}
11:Phase 4: Construct interval

12:

Rtest←𝒟​\[f​\(xtest\)\]R\_\{\\text\{test\}\}\\leftarrow\\mathcal\{D\}\[f\(x\_\{\\text\{test\}\}\)\]
13:

C​\(xtest\)←\[f​\(xtest\)−q^​\(1\+λ​\|Rtest\|\),f​\(xtest\)\+q^​\(1\+λ​\|Rtest\|\)\]C\(x\_\{\\text\{test\}\}\)\\leftarrow\[f\(x\_\{\\text\{test\}\}\)\-\\hat\{q\}\(1\+\\lambda\\left\|R\_\{\\text\{test\}\}\\right\|\),\\;\\;f\(x\_\{\\text\{test\}\}\)\+\\hat\{q\}\(1\+\\lambda\\left\|R\_\{\\text\{test\}\}\\right\|\)\]
14:return

C​\(xtest\)C\(x\_\{\\text\{test\}\}\)

### 4\.3Coordinate\-Aware FNO

We now show that FNO’s architectural symmetry creates a fundamental limitation for PDEs with boundary conditions, and that coordinate channels resolve it\.

###### Proposition 3\(Translation Equivariance of SpectralConv\)\.

For any translationδ\\delta, SpectralConv satisfiesSpectralConv​\(τδ​f\)=τδ​SpectralConv​\(f\)\\text\{SpectralConv\}\(\\tau\_\{\\delta\}f\)=\\tau\_\{\\delta\}\\,\\text\{SpectralConv\}\(f\), where\(τδ​f\)​\(x\):=f​\(x−δ\)\(\\tau\_\{\\delta\}f\)\(x\):=f\(x\-\\delta\)\.

###### Proof\.

By the Fourier shift property,τδ​f^​\(κ\)=e−2​π​i​⟨κ,δ⟩​f^​\(κ\)\\widehat\{\\tau\_\{\\delta\}f\}\(\\kappa\)=e^\{\-2\\pi i\\langle\\kappa,\\delta\\rangle\}\\hat\{f\}\(\\kappa\)\. SinceR​\(κ\)R\(\\kappa\)depends only on frequencyκ\\kappa\(not positionxx\):

SpectralConv​\(τδ​f\)​\(x\)\\displaystyle\\text\{SpectralConv\}\(\\tau\_\{\\delta\}f\)\(x\)=ℱ−1​\[R​\(κ\)​e−2​π​i​⟨κ,δ⟩​f^​\(κ\)\]​\(x\)\\displaystyle=\\mathcal\{F\}^\{\-1\}\\\!\\big\[R\(\\kappa\)\\,e^\{\-2\\pi i\\langle\\kappa,\\delta\\rangle\}\\hat\{f\}\(\\kappa\)\\big\]\(x\)=ℱ−1\[e−2​π​i​⟨κ,δ⟩R\(κ\)f^\(κ\)\]\(x\)=\(τδSpectralConv\(f\)\)\(x\)\.□\\displaystyle=\\mathcal\{F\}^\{\-1\}\\\!\\big\[e^\{\-2\\pi i\\langle\\kappa,\\delta\\rangle\}R\(\\kappa\)\\hat\{f\}\(\\kappa\)\\big\]\(x\)=\(\\tau\_\{\\delta\}\\,\\text\{SpectralConv\}\(f\)\)\(x\)\.\\quad\\square∎

Since the lifting and projection layers are pointwise \(1×11\{\\times\}1convolutions, also equivariant\), the full FNOFFis translation\-equivariant:F​\(τδ​f\)=τδ​F​\(f\)F\(\\tau\_\{\\delta\}f\)=\\tau\_\{\\delta\}\\,F\(f\)\.

###### Proposition 4\(Non\-Equivariance of the Solution Operator\)\.

LetS:k↦uS:k\\mapsto ube the solution operator for a PDE with non\-trivial Dirichlet BCu\|∂Ω=gu\|\_\{\\partial\\Omega\}=g\. ThenSSis not translation\-equivariant: there existδ,k\\delta,ksuch thatS​\(τδ​k\)≠τδ​S​\(k\)S\(\\tau\_\{\\delta\}k\)\\neq\\tau\_\{\\delta\}\\,S\(k\)\.

###### Proof\.

By contradiction\. Takek0≡1k\_\{0\}\\equiv 1\(constant\)\. Thenτδ​k0=k0\\tau\_\{\\delta\}k\_\{0\}=k\_\{0\}, so equivariance requiresS​\(k0\)=τδ​S​\(k0\)S\(k\_\{0\}\)=\\tau\_\{\\delta\}S\(k\_\{0\}\), i\.e\.,u0=S​\(k0\)u\_\{0\}=S\(k\_\{0\}\)is translation\-invariant\. But for non\-trivialgg\(e\.g\.,g​\(x1\)=1−x1g\(x\_\{1\}\)=1\-x\_\{1\}\),u0u\_\{0\}is clearly not translation\-invariant\. Contradiction\.□\\square∎

Propositions[3](https://arxiv.org/html/2609.11935#Thmproposition3)–[4](https://arxiv.org/html/2609.11935#Thmproposition4)establish a structural mismatch: FNO’s hypothesis class \(translation\-equivariant operators\) does not contain the target operatorSS\. This yields:

###### Theorem 1\(Approximation Barrier\)\.

LetFFbe any translation\-equivariant FNO without coordinate channels, andSSthe Darcy solution operator with Dirichlet BCg\|∂Ωg\|\_\{\\partial\\Omega\}\. Then there existsC\>0C\>0\(depending on the PDE operator\) such that

supk‖F​\(k\)−S​\(k\)‖L2≥C⋅‖g−g¯‖L2​\(∂Ω\)\\sup\_\{k\}\\left\\\|F\(k\)\-S\(k\)\\right\\\|\_\{L^\{2\}\}\\geq C\\cdot\\left\\\|g\-\\bar\{g\}\\right\\\|\_\{L^\{2\}\(\\partial\\Omega\)\}whereg¯\\bar\{g\}is the mean ofggon∂Ω\\partial\\Omega\.

###### Proof sketch\.

For constant inputk0≡1k\_\{0\}\\equiv 1, equivariance forcesF​\(k0\)=τδ​F​\(k0\)F\(k\_\{0\}\)=\\tau\_\{\\delta\}F\(k\_\{0\}\)for allδ\\delta, soF​\(k0\)F\(k\_\{0\}\)must be translation\-invariant\. However,S​\(k0\)\|∂Ω=gS\(k\_\{0\}\)\|\_\{\\partial\\Omega\}=g, and‖g−g¯‖\>0\\left\\\|g\-\\bar\{g\}\\right\\\|\>0for non\-uniformgg\. The discrepancy‖F​\(k0\)−S​\(k0\)‖L2​\(∂Ω\)≥C′​‖g−g¯‖\\left\\\|F\(k\_\{0\}\)\-S\(k\_\{0\}\)\\right\\\|\_\{L^\{2\}\(\\partial\\Omega\)\}\\geq C^\{\\prime\}\\left\\\|g\-\\bar\{g\}\\right\\\|propagates to the interior via the PDE’s elliptic regularity, yielding the lower bound\.□\\square∎

###### Theorem 2\(Resolution via Coordinate Channels\)\.

Augmenting the input with coordinate channelsk~=\[k,x1,x2\]\\tilde\{k\}=\[k,x\_\{1\},x\_\{2\}\]breaks translation equivariance, removing the approximation barrier of Theorem[1](https://arxiv.org/html/2609.11935#Thmtheorem1)\.

###### Proof\.

Under a “partial translation”τδk\\tau\_\{\\delta\}^\{k\}\(shiftingkkbut not coordinates\), the augmented input transforms asτδk​k~=\[k​\(x−δ\),x1,x2\]\\tau\_\{\\delta\}^\{k\}\\tilde\{k\}=\[k\(x\-\\delta\),\\,x\_\{1\},\\,x\_\{2\}\]\. Since the coordinate channels remain fixed, this is not a simple translation ofk~\\tilde\{k\}\. SpectralConv mixes the frequency content ofkkwith that of the coordinate channels, producing output that depends on absolute position\. Formally,F~​\(τδk​k~\)≠τδ​F~​\(k~\)\\tilde\{F\}\(\\tau\_\{\\delta\}^\{k\}\\tilde\{k\}\)\\neq\\tau\_\{\\delta\}\\tilde\{F\}\(\\tilde\{k\}\)in general, soF~\\tilde\{F\}is not constrained to the equivariant class\.□\\square∎

#### Group\-theoretic perspective\.

Standard FNO is equivariant to the translation groupG=\(ℝd,\+\)G=\(\\mathbb\{R\}^\{d\},\+\)\. Dirichlet BCs reduce the problem’s symmetry to the trivial group\{e\}\\\{e\\\}\. Coordinate channels encode the group’s orbit parameter \(positionxx\) into the input, effectively lifting the problem to a space where equivariance is no longer a constraint, allowing the network to represent position\-dependent operators\.

#### Summary\.

PI\-CP \(Sections[4\.1](https://arxiv.org/html/2609.11935#S4.SS1)–[4\.2](https://arxiv.org/html/2609.11935#S4.SS2)\) provides the*uncertainty quantification*contribution: intervals with provable coverage and physics\-conditioned spatial adaptivity\. The coordinate\-aware FNO theory \(this subsection\) provides the*architectural*contribution: identifying and resolving a fundamental limitation for Dirichlet BCs\.

#### Why these two contributions are inseparable\.

The connection is PI\-CP\-specific, not a generic “good models enable good UQ” argument\. Without coordinate channels, FNO’s translation equivariance forces a*systematic*error pattern: errors concentrate at boundaries where the equivariance assumption fails \(63% rel\_l2 on Darcy, Theorem[1](https://arxiv.org/html/2609.11935#Thmtheorem1)\)\. The PDE residualR​\(x\)R\(x\)is dominated by this same systematic pattern, making the residual\-error correlation uninformative—both quantities reflect architectural limitation, not genuine prediction difficulty\.

Coordinate channels resolve this by breaking the equivariance constraint, transforming the error structure from*systematic/architectural*to*stochastic/physics\-correlated*\(1% rel\_l2\)\. NowR​\(x\)R\(x\)varies based on local coefficient complexity—exactly the spatially\-varying signal that PI\-CP’s score exploits\. This qualitative transformation is specific to PI\-CP: MC Dropout and CQR do not depend on the residual’s discriminative power—they rely on model\-internal uncertainty or learned quantiles, respectively\.Coordinate channels are therefore a PI\-CP\-specific prerequisite: they ensure the PDE residual is a reliable error proxy, not merely a reflection of architectural bias\.

## 5Experiments

### 5\.1Setup

Scenarios: We evaluate on six physics scenarios spanning three PDE families:

- •*Heat conduction*: 2D \(64×6464\{\\times\}64\) and 3D \(32332^\{3\}\) with spatially\-varying conductivity and heat sources
- •*Structural mechanics*: 2D cantilever beam \(64×6464\{\\times\}64, displacement field\) and 3D \(32332^\{3\}\)
- •*Fluid dynamics*: 2D Darcy flow \(64×6464\{\\times\}64, Matérn GRF permeability\) and 2D Navier\-Stokes Taylor\-Green vortex \(64×6464\{\\times\}64,Re∈\[10,100\]\\text\{Re\}\\in\[10,100\]\)

Model: FNO2d/3d with coordinate channels, modes==12, width==32, layers==4 \(1\.19M parameters\)\. All models trained from scratch with identical hyperparameters across scenarios for fair comparison\.

Training: 1000–2000 samples per scenario, 200–300 epochs, Adam optimizer \(lr=10−3\\text\{lr\}=10^\{\-3\}, weight decay=10−5=10^\{\-5\}\), cosine annealing schedule\. Data split: 1000 train / 200 calibration / 300 test\. All experiments on NVIDIA V100 \(16GB\)\.

UQ Configuration:α=0\.1\\alpha=0\.1\(90% nominal coverage\)\. Conformal calibration on 200 held\-out samples\. MC Dropout: 30 forward passes with dropout==0\.1\. Deep Ensemble: 3 independently trained models\. PDE residuals computed via finite differences \(thermal\), finite volume \(Darcy\), and spectral \(NS\)\. Empirical coverage on 300 test samples has theoretical standard deviation±\\pm1\.7% \(Binomial\(300,0\.90\)\(300,0\.90\)\), so inter\-method coverage differences within±\\pm2% are not statistically significant\.

### 5\.2Coordinate Channel Ablation

Figure[1](https://arxiv.org/html/2609.11935#S5.F1)and Table[1](https://arxiv.org/html/2609.11935#S5.T1)show the effect of coordinate channels across five scenarios\.

![Refer to caption](https://arxiv.org/html/2609.11935v1/x1.png)Figure 1:Coordinate channel ablation across 5 scenarios\. Without coordinates, FNO’s translation equivariance prevents learning Dirichlet boundary conditions, with errors up to 76%\. Adding coordinate channels\[x,y\]\[x,y\]reduces errors by1\.21\.2–63×63\\times\.Table 1:Coordinate channel ablation: rel\_l2 \(%\) with vs without grid coordinates\.
### 5\.3PI\-CP Results on Darcy Flow

Figure[2](https://arxiv.org/html/2609.11935#S5.F2)illustrates the spatial adaptivity of PI\-CP intervals compared to standard CP\.

![Refer to caption](https://arxiv.org/html/2609.11935v1/x2.png)Figure 2:PI\-CP on Darcy flow: coverage vs\. spatial adaptivity\. All methods maintain∼\\sim90% coverage\. PI\-CP \(λ=1\\lambda\{=\}1\) achieves width CV==0\.221 \(spatially adaptive\) with only 4% width increase over standard CP\.Table 2:PI\-CP comparison on Darcy flow \(model rel\_l2==1\.0%\)\. Width CV measures spatial adaptivity\.#### Analysis\.

All methods achieve the target coverage \(89\.489\.4–89\.7%89\.7\\%\), confirming the distribution\-free guarantee\. The key differentiator is spatial adaptivity, measured by the width coefficient of variation \(CV\): standard and normalized CP produce uniform\-width intervals \(CV≈0\\approx 0\), while PI\-CP achieves CV==0\.221 atλ=1\\lambda\{=\}1with only4%4\\%average width increase\. Theλ\\lambdaparameter provides smooth control: atλ=5\\lambda\{=\}5, the adaptivity increases \(CV==0\.643\) at the cost of31%31\\%wider intervals\. PI\-Normalized CP combines physics and epistemic uncertainty, achieving comparable adaptivity to PI\-CP with slightly tighter intervals\.

### 5\.4Architecture Ablation

Table 3:FNO architecture ablation on Darcy flow \(1800 train, 300 epochs\)\. rel\_l2 in %\.Finding: Diminishing returns across all hyperparameters\. Darcy flow at64×6464\{\\times\}64is dominated by low\-frequency content, so even small models \(m=8m\{=\}8,w=32w\{=\}32,L=4L\{=\}4\) achieve<0\.5%<0\.5\\%rel\_l2\. The sample scaling follows an approximate power law:rel\_l2∝N−0\.5\\text\{rel\\\_l2\}\\propto N^\{\-0\.5\}, whereNNis the training set size\. Doubling the samples from 1000 to 2000 reduces error from 0\.66% to 0\.45%\. Figure[3](https://arxiv.org/html/2609.11935#S5.F3)visualizes the modes scaling\.

![Refer to caption](https://arxiv.org/html/2609.11935v1/x3.png)Figure 3:FNO Fourier modes ablation on Darcy flow\. Error plateaus atm=8m\{=\}8, confirming low\-frequency dominance\.
### 5\.5Baseline Comparison

Figure[4](https://arxiv.org/html/2609.11935#S5.F4)compares FNO against three baselines\.

![Refer to caption](https://arxiv.org/html/2609.11935v1/x4.png)Figure 4:Architecture comparison on Darcy flow\. FNO \(0\.47%0\.47\\%\) outperforms CNN \(9\.8×9\.8\\times\), DeepONet \(11\.5×11\.5\\times\), and MGN \(13\.7×13\.7\\times\)\.Table 4:Architecture comparison on Darcy flow \(1800 train, 300 epochs\)\.†MGN evaluated at32×3232\{\\times\}32due to graph construction cost; all others at64×6464\{\\times\}64\.#### Analysis\.

FNO outperforms all baselines by9\.89\.8–13\.7×13\.7\\times\. CNN’s local receptive field \(5×55\{\\times\}5convolutions\) cannot capture the global dependencies inherent in elliptic PDEs like Darcy flow\. DeepONet’s branch\-trunk decomposition struggles with the spatial structure of the permeability\-to\-pressure mapping\. Note that MGN was evaluated at32×3232\{\\times\}32\(vs\.64×6464\{\\times\}64for all others\) due to graph construction cost, so its13\.7×13\.7\\timesdisadvantage is partly attributable to lower resolution; the fair\-gap architecture comparison is FNO vs\. CNN/DeepONet \(9\.89\.8–11\.5×11\.5\\times\)\. FNO’s spectral convolution captures global correlations inO​\(N​log⁡N\)O\(N\\log N\)time, explaining its superiority for smooth PDE solutions\.

### 5\.6Navier\-Stokes: Taylor\-Green Vortex

Table 5:Taylor\-Green vortex: FNO2d with Re∈\[10,100\]\\in\[10,100\], 1500 samples, 300 epochs\.#### Navier\-Stokes result\.

The Taylor\-Green vortex atRe∈\[10,100\]\\text\{Re\}\\in\[10,100\]is well\-resolved with0\.31%0\.31\\%test error, confirming that FNO can handle advection\-diffusion dynamics\. The vorticity fieldω\\omegais predicted from the initial conditionω0\\omega\_\{0\}and viscosityν\\nu, with coordinate channels providing spatial anchoring\.

### 5\.7Burgers Equation: Limitation with Shocks

To probe FNO’s limitations, we evaluate on the 1D viscous Burgers equation∂tu\+u​∂xu=ν​∂x​xu\\partial\_\{t\}u\+u\\partial\_\{x\}u=\\nu\\partial\_\{xx\}u, whose solutions develop shocks at low viscosityν\\nu\. Data is generated via the Cole\-Hopf transformation\(Cole,[1951](https://arxiv.org/html/2609.11935#bib.bib17)\)with random multi\-mode sinusoidal initial conditions\. We test two regimes: smooth \(ν∈\[0\.05,0\.5\]\\nu\\in\[0\.05,0\.5\]\) and shock\-forming \(ν∈\[0\.001,0\.02\]\\nu\\in\[0\.001,0\.02\]\)\.

Table 6:Burgers equation: FNO accuracy in smooth vs\. shock\-forming regimes\. 1000 training samples, 200 epochs, V100 GPU,64×1664\\times 16grid \(1D replicated via Cole\-Hopf\)\.#### Result\.

FNO achieves5\.02%5\.02\\%error on smooth Burgers—higher than steady\-state PDEs \(0\.030\.03–0\.33%0\.33\\%\) but reasonable for a time\-evolution problem\. However, in the shock\-forming regime, FNO completely fails \(\>100%\>100\\%rel\_l2\)\. This is a*known limitation*of spectral neural operators: discontinuous solutions require infinitely many Fourier modes, but FNO truncates tommmodes, causing Gibbs phenomena\. This motivates future work on wavelet\-based or multi\-resolution architectures\(Kovachki et al\.,[2023](https://arxiv.org/html/2609.11935#bib.bib13)\)\.

### 5\.8UQ Cross\-Scenario Comparison

Table[7](https://arxiv.org/html/2609.11935#S5.T7)compares all six UQ methods across four scenarios \(the 2D scenarios with computed PDE residuals\)\. This is the most comprehensive UQ comparison for neural operators to date\.

Table 7:UQ cross\-scenario comparison: coverage and spatial adaptivity at 90% nominal level\. Conformal methods \(Std CP, Norm CP, PI\-CP, CQR\) satisfy the distribution\-free guarantee; MC Dropout and Deep Ensembles do not\. Width CV measures spatial adaptivity \(higher = more adaptive\)\.#### Analysis\.

Five key findings emerge: \(1\)Coverage guarantee holds for all Conformal methods: Std CP, Norm CP, PI\-CP, and CQR all achieve 90–91% coverage—within1​σ1\\sigma\(±\\pm1\.7%\) of the 90% nominal level, confirming that inter\-method coverage differences are statistically indistinguishable, as conformal theory predicts\. \(2\)Adaptivity is scenario\-dependent for both PI\-CP and CQR: PI\-CP achieves strong spatial adaptivity on NS \(CV==0\.493\) and Darcy \(CV==0\.233\) where PDE residuals correlate with errors, but degrades on cantilever \(CV==0\.001, near\-zero residual\-error correlation\) and over\-widens on thermal \(CV==1\.062,ρ=−0\.14\\rho\{=\}\{\-\}0\.14\)\. CQR shows a complementary pattern: strong on Darcy \(CV==0\.543\) and NS \(CV==0\.465\), but collapses on cantilever \(CV≈0\\approx 0\)\. Neither method dominates universally—the choice depends on the scenario’s residual\-error structure\. \(3\)Non\-conformal methods are unstable: MC Dropout ranges from 82\.2% to 100%, and Deep Ensembles from 82\.1% to 97\.8%, providing no reliability contract\. \(4\)CQR requires retraininga quantile model \(∼\\sim400s per scenario on V100\), while PI\-CP works on any pre\-trained point\-prediction model with zero retraining cost\. \(5\)PI\-CP’s key practical advantage is deployment simplicity: guaranteed coverage with no retraining, moderate adaptivity whenρ\>0\\rho\>0, and a clear diagnostic \(ρ\\rho\) for predicting effectiveness\.

### 5\.9Summary of Physics Scenarios

Table 8:All physics scenarios: accuracy and coverage summary\.![Refer to caption](https://arxiv.org/html/2609.11935v1/x5.png)Figure 5:RelativeL2L^\{2\}error across all six physics scenarios\. All achieve<2%<2\\%error with coordinate\-aware FNO\.

## 6Discussion

#### When PI\-CP Helps\.

PI\-CP is most effective when PDE residuals are dominated by*model error*rather than discretization error, and when the residual has spatial structure correlated with actual errors\. We recommend computing the Pearson correlationρ=corr​\(\|R​\(x\)\|,\|y−f​\(x\)\|\)\\rho=\\text\{corr\}\(\\left\|R\(x\)\\right\|,\\left\|y\-f\(x\)\\right\|\)on a validation set as a diagnostic: ifρ\>0\\rho\>0, PI\-CP will tighten intervals where physics is well\-satisfied; ifρ≤0\\rho\\leq 0, PI\-CP may over\-widen\.

Empirically, PI\-CP achieves strong spatial adaptivity \(width CV==0\.23–0\.49\) on Darcy flow and Navier\-Stokes, where PDE residuals \(finite\-volume vorticity transport\) correlate well with model errors\. In contrast, the cantilever elasticity residual shows near\-zero adaptivity \(CV==0\.001\) due to poor residual\-error correlation, and thermal 2D \(finite\-difference Laplacian\) showsρ=−0\.14\\rho=\-0\.14, causing over\-widening\.

#### Theλ\\lambdaTradeoff\.

The hyperparameterλ\\lambdacontrols physics weighting:λ→0\\lambda\\to 0recovers standard CP \(no adaptivity\); largeλ\\lambdaproduces fully adaptive but potentially wider intervals\. We recommendλ∈\[0\.5,2\.0\]\\lambda\\in\[0\.5,2\.0\], selected via validation set width minimization subject to coverage constraint\.

#### Coordinate Channels as Prerequisite\.

For PDEs with Dirichlet BCs, coordinate channels are*essential*, not optional \(Theorem[1](https://arxiv.org/html/2609.11935#Thmtheorem1)\)\. Without them, model error is dominated by the approximation barrier \(63% on Darcy\), making any UQ method meaningless\.Coordinate channels should be the default for FNO applied to bounded domains\.

#### Limitations\.

\(1\) Theorem[1](https://arxiv.org/html/2609.11935#Thmtheorem1)provides an existence bound; the exact constantCCdepends on the PDE operator and domain geometry\. \(2\) PI\-CP’s effectiveness depends on residual\-error correlation, which varies by scenario and discretization scheme\. \(3\) We analyze Dirichlet BCs; Neumann/Robin BCs have similar symmetry\-breaking but require separate analysis\. \(4\) FNO fails on shock\-forming Burgers \(\>100%\>100\\%error atν<0\.02\\nu<0\.02\), a known limitation of spectral methods for discontinuous solutions \(§[5](https://arxiv.org/html/2609.11935#S5), Table[6](https://arxiv.org/html/2609.11935#S5.T6)\)\. This limitation propagates to PI\-CP: without an accurate base model, the PDE residual is not a meaningful error indicator\. \(5\) Our exchangeability assumption may be violated under distribution shift; weighted conformal prediction\(Barber et al\.,[2023](https://arxiv.org/html/2609.11935#bib.bib19)\)and adaptive conformal inference\(Gibbs & Candès,[2021](https://arxiv.org/html/2609.11935#bib.bib12)\)can address this, which we leave for future work\.

## 7Conclusion

We presented Physics\-Informed Conformal Prediction \(PI\-CP\), a framework that bridges conformal prediction and physics\-informed machine learning by embedding PDE residuals into the nonconformity score\. PI\-CP provides provable, distribution\-free coverage guarantees—identical to standard CP in the marginal sense—while offering spatial adaptivity when the PDE residual serves as a reliable error proxy\. Across six physics scenarios, all four Conformal methods \(Standard, Normalized, PI\-CP, CQR\) achieve 89–91% coverage, while MC Dropout and Deep Ensembles exhibit unstable coverage \(82–100%\) without guarantees\.

A key practical finding is that PI\-CP’s effectiveness depends on the correlation between PDE residuals and actual model errors\. When the residual is dominated by model error \(as in Darcy flow\), PI\-CP achieves spatial adaptivity with minimal width increase \(\+4%\)\. When discretization error dominates \(as in thermal\), the residual\-error correlation breaks down, and PI\-CP may over\-widen\. We recommend computing the Pearsonρ​\(\|R\|,\|y−f\|\)\\rho\(\|R\|,\|y\-f\|\)as a pre\-application diagnostic\.

Our second contribution—the coordinate\-aware FNO theory—addresses a fundamental architectural limitation\. We proved that FNO’s translation equivariance creates an approximation barrier for Dirichlet BCs, and showed that coordinate channels resolve this with up to63×63\\timesimprovement\. This finding suggests that the symmetry structure of neural operators should be carefully matched to the symmetry structure of the target PDE operators, particularly regarding boundary conditions\.

Future work includes: \(1\) extending PI\-CP to temporal/spatiotemporal settings with adaptive conformal inference\(Gibbs & Candès,[2021](https://arxiv.org/html/2609.11935#bib.bib12)\), \(2\) analyzing the optimalλ\\lambdaselection theoretically, and \(3\) validating on industrial\-scale problems with real CAE data \(e\.g\., Ansys/Abaqus simulations of aircraft components or reactor thermal\-hydraulics\), rather than the synthetic/analytical benchmarks used here\. Our experiments use analytically generated data \(Cole\-Hopf, Taylor\-Green, Matérn GRF\) and FEM simulations with controlled parameters; real engineering datasets introduce mesh irregularity, multi\-physics coupling, and measurement noise that may affect PI\-CP’s residual\-error correlation\. Addressing these is essential for deployment in safety\-critical applications\.

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## Appendix AReproducibility Details for UQ Experiments

Table[7](https://arxiv.org/html/2609.11935#S5.T7)\(reproduced in the main text, §[5](https://arxiv.org/html/2609.11935#S5)\) reports the full UQ cross\-scenario comparison\. Here we provide additional reproducibility details\.

All Conformal methods use a calibration set of 200 held\-out samples and a test set of 300 samples per scenario \(CQR: 250 calibration / 250 test\)\. The significance level isα=0\.1\\alpha=0\.1\(90% nominal coverage\)\. MC Dropout uses 30 stochastic forward passes with dropout rate 0\.1 \(applied in each Fourier layer\)\. Deep Ensemble uses 3 independently trained FNO models with different random seeds\. CQR uses an FNO with out\_channels==3 trained with pinball loss at quantiles\[0\.05,0\.5,0\.95\]\[0\.05,0\.5,0\.95\]for 200 epochs\. PDE residuals are computed via: finite differences \(thermal 2D\), finite element residual \(cantilever 2D\), finite volume \(Darcy\), and spectral differentiation \(NS Taylor\-Green\)\. The physics weighting isλ=1\.0\\lambda=1\.0for all PI\-CP results\.

## Appendix BImplementation Details

The PhysAI platform is implemented in Python 3\.10\+ with PyTorch\. All experiments use the same FNO architecture \(modes==12, width==32, layers==4,∼\\sim1\.2M parameters\) for cross\-scenario consistency\. Key components:

- •Models: FNO2d/3d with spectral convolution
- •UQ: ConformalPredictor with 4 methods \(Standard, Normalized, PI\-CP, PI\-Normalized\), MC Dropout, Deep Ensemble, CQR
- •Physics solvers: Darcy \(sparse direct\), Navier\-Stokes \(Taylor\-Green analytical\), Heat \(finite difference\), Structural \(FEM\)

Training performed on NVIDIA V100 \(16GB\)\. Code will be released upon publication\.