neural-operators

Proposes REEF-GP, a post-hoc uncertainty quantification framework that fits a Gaussian process to the residuals of a frozen neural operator using its internal embeddings, enabling geometry-aware and calibrated uncertainties at low cost.

Operator Boosting is a stagewise residual-learning framework that constructs compact neural operator surrogates for PDEs by training tiny models on residual fields. It achieves accuracy comparable to or better than full-size models while reducing parameters by up to 95%, demonstrating Pareto improvements on several benchmarks.

Anima Anandkumar highlights that neural operators, despite simple benchmarks, have achieved massive speedups (10,000–million times) in hard real-world problems like high-resolution AI weather modeling (FourCastNet) and nuclear fusion turbulence, referencing a new paper showing learned solvers become more cost-effective as PDE tasks get harder.

This paper introduces Topological Neural Operators, which lift neural operators from point-only domains to cell complexes, embedding geometry and topology to reduce the learning burden. It demonstrates that operator learning improves when geometry is not an afterthought, though the topology remains prescribed.

Proposes the first application of split conformal prediction to neural operator-based physics simulation, providing distribution-free prediction intervals with finite-sample coverage guarantees and adaptive-width intervals using MC Dropout uncertainty.

Fourier neural operators (FNOs) achieve extrapolation success in modeling periodically driven quantum systems, capturing temporal correlations in frequency space for physically faithful dynamics beyond training data.

Proposes semigroup consistency as a diagnostic for evaluating learned physics simulators, showing that normalized semigroup error correlates with rollout degradation in heat and Burgers dynamics using ConvNet and FNO baselines.

Investigates neural integral-operator-based models for fMRI encoding and decoding tasks, focusing on the role of nonlocal spatiotemporal context and showing that larger temporal windows improve performance across datasets.

This paper proves the first universal approximation theorems for nonlinear operators and their derivatives in infinite-dimensional settings, extending classical results to operator learning architectures like DeepONet and PCA-Net.

This paper investigates the generalization behavior of Fourier Neural Operators and Deep Operator Networks under distribution shifts in a variable-coefficient wave equation, revealing that FNO struggles with high-frequency inputs while DeepONet shows milder degradation.

This paper establishes quantitative Sobolev approximation bounds for neural operators, proving that operators can be uniformly approximated with explicit complexity-error relations. It validates these theoretical bounds using Fourier Neural Operators on the Burgers' equation, demonstrating that Sobolev-space approximation theory accurately predicts scaling behavior.

This paper proposes a new architecture that augments Flux Neural Operators with recurrent Vision Transformers to solve conservation laws as a foundation model. It demonstrates robust generalization and long-time prediction capabilities across diverse conservative systems without explicit access to governing equations.