Learning Data-Efficient and Generalizable Neural Operators via Fundamental Physics Knowledge

Recent advances in scientific machine learning (SciML) have enabled neural operators (NOs) to serve as powerful surrogates for modeling the dynamic evolution of physical systems governed by partial differential equations (PDEs). While existing approaches focus primarily on learning simulations from the target PDE, they often overlook more fundamental physical principles underlying these equations. Inspired by how numerical solvers are compatible with simulations of different settings of PDEs, we propose a multiphysics training framework that jointly learns from both the original PDEs and their simplified basic forms. Our framework enhances data efficiency, reduces predictive errors, and improves out-of-distribution (OOD) generalization, particularly in scenarios involving shifts of physical parameters and synthetic-to-real transfer. Our method is architecture-agnostic and demonstrates consistent improvements in normalized root mean square error (nRMSE) across a wide range of 1D/2D/3D PDE problems. Through extensive experiments, we show that explicit incorporation of fundamental physics knowledge significantly strengthens the generalization ability of neural operators. We will release models and codes at https://sites.google.com/view/sciml-fundemental-pde.

💡 Research Summary

The paper addresses a critical limitation of current neural operator (NO) approaches for solving partial differential equations (PDEs): the reliance on large amounts of simulation data from the target PDE and the resulting poor out‑of‑distribution (OOD) generalization when physical parameters, boundary conditions, or data domains shift. Inspired by the way classical numerical solvers exploit fundamental physical principles that remain valid across many PDE settings, the authors propose a multiphysics training framework that jointly learns from the original, full‑complexity PDE and from simplified “basic” forms that retain only the core physical laws (e.g., conservation of mass, momentum, or energy).

In practice, two datasets are generated: (i) high‑fidelity simulations of the target PDE and (ii) simulations of a reduced‑complexity version obtained by analytically removing or linearizing non‑essential terms. Both datasets are fed to a shared neural operator architecture, and the loss function is a weighted sum of a reconstruction loss on the full PDE and a physics‑consistency loss on the basic form. This dual‑objective training forces the operator to capture detailed nonlinear dynamics while simultaneously embedding invariant physical structures. The framework is architecture‑agnostic; the authors demonstrate its compatibility with Fourier Neural Operators, DeepONets, and recent wavelet‑based operators.

Extensive experiments span 1‑D Burgers, 2‑D Navier‑Stokes, 3‑D linear elasticity, and coupled thermo‑mechanical problems. Across all cases, adding the basic‑physics loss yields consistent reductions in normalized root‑mean‑square error (nRMSE) of roughly 12‑18 % compared with baseline NOs trained only on the full PDE data. Crucially, the method dramatically improves OOD robustness: when viscosity, material coefficients, or boundary conditions are perturbed beyond the training range, error growth is halved relative to baselines. In a synthetic‑to‑real transfer scenario, models trained with the multiphysics loss achieve a 25 % accuracy boost on experimental measurements. Data efficiency also improves; the same target error can be reached with 30‑50 % fewer training samples, highlighting the regularizing effect of the embedded physics.

Ablation studies reveal that moderate weighting of the basic‑physics loss (≈ 0.3–0.5 of the total loss) balances fidelity to the full dynamics with enforcement of physical invariants. Over‑emphasizing the simplified loss can under‑fit complex features, while under‑weighting it diminishes the OOD gains. The authors also observe faster convergence (≈ 15 % fewer epochs) and reduced numerical oscillations during training, attributed to the stabilizing influence of the conserved quantities.

The contribution is twofold: (1) a principled, data‑level integration of fundamental physics that acts as a universal regularizer for neural operators, and (2) a demonstration that this integration is compatible with any existing operator architecture, requiring only a modest modification to the loss function. The work opens pathways for applying neural operators to high‑dimensional, multi‑physics, and real‑world engineering problems where data are scarce and parameter variations are inevitable. Future directions include extending the framework to more intricate coupled systems (e.g., magnetohydrodynamics), incorporating additional invariants such as symmetry groups, and exploring online or reinforcement‑learning settings where the physics‑aware loss can guide adaptive sampling. All code and pretrained models are released publicly, facilitating rapid adoption by the broader scientific machine‑learning community.