Learning Data-Efficient and Generalizable Neural Operators via Fundamental Physics Knowledge
Siying Ma, Mehrdad M. Zadeh, Mauricio Soroco, Wuyang Chen, Jiguo Cao, Vijay Ganesh
TL;DR
This paper introduces a physics-informed training framework for neural operators that improves data efficiency, accuracy, and out-of-distribution generalization by leveraging fundamental physical principles and simplified PDE forms.
Contribution
The proposed multiphysics training approach integrates basic physics knowledge into neural operator learning, enhancing generalization and transferability across diverse PDE problems.
Findings
Improved normalized root mean square error (nRMSE) across multiple PDE benchmarks.
Enhanced out-of-distribution generalization for physical parameter shifts.
Architecture-agnostic framework applicable to 1D/2D/3D PDEs.
Abstract
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…
Peer Reviews
Decision·ICLR 2026 Poster
- The paper studies an important but often overlooked probably on efficient data generation for neural operators. - When comparing against the baseline, the paper controls the overall run time budget. - The experiment show consistent improvement.
- It is not always obviously which is the best basic form to each target equation. I don't think there is a canonical choice, and the performance depends on the choise of augmented basic PDE. For example on 2D NS the treatment is more significant, but not as much on KS. It would be better to add some ablations to justify the choice. - Overall I like this paper but I don't like the storytelling. The main message should be "it is helpful to generate additional data in simpler form". It is a bit sp
The authors target two central SciML issues, 1. data hunger and 2. poor OOD transfer, for operator learning across 1D/2D/3D PDEs (Diffusion-Reaction, Navier–Stokes, Kuramoto–Sivashinsky, plus ScalarFlow). The key contribution lies in identifying and leveraging "fundamental physics knowledge" through decomposed basic PDE forms. This has not been explored extensively in the neural operator literature. Proposed benefits such as: Data efficiency, Long horizon stability , OOD generalization , ar
The term "fundamental physics knowledge" is somewhat vague and could be better defined Section 3.1 could be more systematic in explaining the decomposition principles Some notation inconsistencies (e.g., switching between v and u for solutions Missing error bars in main results (added later in appendix) there is limited statistical analysis The ScalarFlow experiment (Section 4.5) is somewhat disconnected and brief Claims about "fundamental physics knowledge" being key are not fully validated (c
- The proposed method is well-motivated. The authors provide a critical observation by evaluating existing SciML foundation models. They find a strong correlation between a model's performance on the original PDE and its performance on the fundamental components of that PDE (e.g., pure diffusion for a reaction-diffusion system). However, the absolute error on these basic terms remains high, indicating that even powerful models lack a robust understanding of the foundational physics, which motiva
- Heuristic Nature of Decomposition: The process for selecting terms for the "basic form," while physically intuitive, remains heuristic. A more formalized principle or an ablation study discussing the impact of alternative decompositions more prominently would strengthen the methodology. - Inadequate Mechanistic Explanation for the Efficacy of Basic Form Data: A significant weakness of the paper lies in its insufficient exploration of the underlying mechanisms by which the "basic form" data
Code & Models
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsModel Reduction and Neural Networks · Machine Learning in Materials Science · Generative Adversarial Networks and Image Synthesis