pde

This paper introduces topology as a distinct generalization axis for neural operators in PDE solving, using Hodge heat flow and the TopoBox-3D framework to show its influence across the Hodge spectrum beyond geometric compatibility.

SS-eSOAP introduces a self-scaled adaptive preconditioning method for physics-informed neural networks, improving optimization by reducing residuals and achieving higher accuracy on various PDE benchmarks compared to baseline methods.

This paper presents a method to rank neural operator models during deployment using shared physics responses, achieving high accuracy without ground-truth reference solutions for scientific computing applications.

Presents GeoIncNO, a geometry-aware incremental neural operator that improves long-horizon PDE prediction via residual latent increments and mean-fluctuation decoupled reconstruction, achieving better stability and spectral fidelity on 1D/2D/3D benchmarks.

This paper introduces Neural Networks with Local Converging Inputs (NNLCI) to accelerate PDE-based option pricing, combining coarse and refined mesh solutions to reduce RMSE by 4–12x in multi-asset Black–Scholes and Heston models.

EvoPINN is an agentic framework that reformulates PINN development as an execution-grounded algorithm discovery problem, using an LLM agent to propose programmatic modifications. It autonomously discovers PDE-specialized learning algorithms, including a novel architecture called SLRC-PINN, which outperforms baselines across diverse PDE regimes.

A technical blog post introducing Walk on Decomposed Subdomains (WODS), a grid-free Monte Carlo method for solving elliptic PDEs with complex geometries, contrasted with finite differences and finite elements.

Introduces latent PDE mapping, a physics-informed learning technique that enables efficient geometric generalization with sparse training data by pulling back PDE residuals to a latent geometry. Demonstrates significant error reduction on cardiac electrophysiology PDE benchmarks using only 15 training samples.

SechKAN is a novel Kolmogorov-Arnold Network architecture that uses hyperbolic secant functions as basis functions, achieving competitive performance in function fitting, PDE problems, and image classification tasks while maintaining parameter efficiency comparable to MLPs.

This paper presents an agentic AI scientific community of virtual labs that autonomously discover neural operator architectures for PDE problems. Using LLM planners, numerical workers, and reviewers under a citation-based economy, the system produces high-accuracy hybrid architectures, with results suggesting no universal winner among operator families.

This paper introduces LiNO, a neural operator that uses a lifting-based multiresolution decomposition to learn solution operators for PDEs. It demonstrates strong performance on benchmarks including Darcy flow, Poisson equation, and Navier-Stokes, capturing both global dynamics and fine-scale structure.

This paper presents a geometry-conditioned Fourier Neural Operator (FNO) to learn the solution operator for the cubic nonlinear Schrödinger equation on periodic domains with varying aspect ratios. Numerical experiments show the model captures distinct Sobolev norm behaviors on rational and irrational tori, demonstrating geometry-aware neural operators for dispersive PDEs.

Introduces the ICML 2026 paper Functional Attention, which treats functions as first-class citizens and replaces softmax point-to-point similarity with structured linear operators. It addresses issues of discretization, resolution sensitivity, and high computational complexity in traditional Transformers when handling continuous functions. Achieves or surpasses SOTA in tasks like PDE solving and 3D segmentation, and exhibits strong OOD generalization.

This paper provides approximation and generalization error estimates for multi-input neural operators measured in Sobolev norms, analyzing how multiple input functions with different domains and regularities affect error bounds, applicable to PDE and scientific computing problems.

SirenFNO leverages sinusoidal representation networks to learn full-frequency Fourier kernels, eliminating frequency truncation and achieving significant parameter reductions while improving accuracy on PDE benchmarks.

LFNO is a unified neural operator framework that integrates Laplace and Fourier transforms to decompose system dynamics into transient and steady-state components, significantly outperforming existing operators on ODE and PDE benchmarks.

This paper develops a PAC-Bayesian framework for physics-informed machine learning, providing high-probability generalization guarantees for unbounded losses. It proposes a multi-task perspective that jointly handles data fidelity, PDE residuals, and boundary conditions, and introduces a self-bounding learning algorithm.

This paper identifies neural network training as a search through Hamilton-Jacobi initial-value problems, showing that residual networks, transformers, and RNNs discretize the same class of viscous Hamilton-Jacobi equations. It derives quantitative consequences including minimax optimal generalization rates, adversarial robustness bounds, and a closed-form influence function.