Learning Physical Operators using Neural Operators

2026-02-26

AI Summary

This work proposes a modular mixture-of-experts architecture for solving partial differential equations, addressing the limited generalization and reliance on fixed temporal discretizations of existing neural operators. By integrating an operator splitting strategy, the approach explicitly encodes physical structure: nonlinear physics components are learned via neural operators, while linear parts are approximated using fixed finite-difference convolutions. The entire framework is embedded within a Neural ODE formulation, enabling continuous-time prediction. This design yields superior out-of-distribution generalization, higher parameter efficiency, and faster convergence on both incompressible and compressible Navier–Stokes equations. Notably, the method supports temporal extrapolation beyond the training time horizon, demonstrating robust predictive capability in unseen regimes.

Technology Category

• Machine Learning: Mixture of Experts (MoE)
• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Architectures and workflows that use LLMs for crowd work
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work introduces a physics-informed training framework that addresses these limitations by decomposing PDEs using operator splitting methods, training separate neural operators to learn individual non-linear physical operators while approximating linear operators with fixed finite-difference convolutions. This modular mixture-of-experts architecture enables generalisation to novel physical regimes by explicitly encoding the underlying operator structure. We formulate the modelling task as a neural ordinary differential equation (ODE) where these learned operators constitute the right-hand side, enabling continuous-in-time predictions through standard ODE solvers and implicitly enforcing PDE constraints. Demonstrated on incompressible and compressible Navier-Stokes equations, our approach achieves better convergence and superior performance when generalising to unseen physics. The method remains parameter-efficient, enabling temporal extrapolation beyond training horizons, and provides interpretable components whose behaviour can be verified against known physics.

Problem

• neural operators
• generalization
• partial differential equations
• temporal discretisation
• physics-informed learning

Innovation

• neural operators
• physics-informed learning
• operator splitting
• neural ODE
• PDE surrogates