Learning Physical Operators using Neural Operators

TL;DR

This work introduces a physics-informed training framework that addresses PDE 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.

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.

Figures (44)

• Figure 1: Comparison of operator learning approaches for PDE solving. The figure contrasts three methods: (top left) the traditional autoregressive approach where a single neural operator learns the solution operator mapping directly from state $u^n$ to $u^{n+1}$; (bottom left) the neural ODE approach where a neural operator learns the dynamics operator $\dv{u}{t}$ integrated with an ODE solver; and (right) the proposed OpsSplit method where individual neural operators learn specific physical operators $(\nabla \cross, \nabla \cdot)$ and are combined via operator splitting to compute $\dv{u}{t}$ which is integrated using an ODE solver. The OpsSplit approach decomposes the PDE into its constituent physical operators, with linear operators approximated by convolutions and non-linear operators learned by neural networks, enabling physics-informed and modular PDE solving.
• Figure 2: Rollout error for incompressible Navier--Stokes equations. FNO (\ref{['fig:nrmse_incomp_100_iid_fno', 'fig:nrmse_incomp_100_ood_fno']}) and CNO (\ref{['fig:nrmse_incomp_100_iid_cno', 'fig:nrmse_incomp_100_ood_cno']}) predictive error growth for in-distribution and out-of-distribution cases. The temporal extrapolation region is shaded orange. OpsSplit accumulates fewer errors and provides more stable temporal rollout than autoregressive and neural ODE methods across architectures and scenarios.
• Figure 3: Neural operator learned convection (\ref{['fig:convops_neural']}) versus numerical method. NO captures convection nuances and advection-driven flow trends. Qualitative interpretability study: learned convection exists in latent space; numerical convection shown in physical space, normalised to [-1, 1].
• Figure 4: Continuity equation (\ref{['eq:ns_cont']}) violation across FNO predictions for OOD modeling.
• Figure 5: Rollout error for the compressible Navier--Stokes equations. Similar to \ref{['fig:incomp_rollout_error']}, show the rollout error of various methods for an FNO and U-Net for both in and out-of-distribution scenarios. In most cases, across neural operator architectures, our method of deploying operator splitting to learn the physical operator accumulates less error and provides a more stable temporal rollout than autoregressive and neural ODE-based methods.

Theorems & Definitions (2)

• Theorem 1: Generalisation Error under Parameter Shift
• proof