operator-learning

Proposes KL-DNN, a scalable operator learning framework that uses Karhunen-Loève expansions to handle large-scale PDE problems, achieving lower errors and two-order-of-magnitude speedup over DeepONet on a 3D carbon storage problem.

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.

Proposes a hierarchical attention mechanism using overlapping Schwarz domain decomposition to replace dense global low-rank attention with a two-level additive structure of local and coarse blocks, showing faster training and better accuracy with fewer parameters.

Physics-conforming Latent Twins is a framework for learning latent surrogate solution operators that enforce physical principles such as conservation laws and dissipative inequalities by design, using a constraint-transfer approach and structure-preserving latent dynamics.

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.

DeepMDMD combines deep learning with algebraic constraints to learn compact, dynamically coherent Koopman operator representations that enforce the product rule as an exact constraint. The method outperforms geometric approaches on high-dimensional chaotic and fluid dynamics problems, reducing spectral pollution and enabling stable long-term forecasting.

Functional Attention is a novel attention mechanism that reinterprets attention as a functional correspondence between adaptive bases, replacing softmax affinities with structured linear operators inspired by geometric functional maps. The method achieves state-of-the-art performance on operator learning tasks including PDE solving and 3D segmentation while remaining resolution-invariant.

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.

Introduces UFO, a cross-domain neural operator framework that adaptively learns operators across different representational domains, enabling discretization-decoupled predictions robust to distribution shifts.

This paper proposes a topology-preserving neural operator learning method using Hodge decomposition to separate topological and geometric components, improving accuracy and efficiency on geometric meshes.