Data-Efficient Neural Operator Training via Physics-Based Active Learning

2026-05-20

AI Summary

This work addresses the high data requirements and computational cost of neural operators for solving partial differential equations (PDEs) by introducing a physics-informed active learning framework. For the first time, the PDE residual is employed as an acquisition function to guide the selection of the most informative training samples in regions where the model’s physical understanding is weakest, thereby injecting a strong physics-based inductive bias. By synergistically integrating neural operators, active learning, and physics-informed machine learning, the proposed method substantially improves data efficiency. Experiments on the one-dimensional Burgers equation and the two-dimensional compressible Navier–Stokes equations demonstrate clear superiority over random sampling, achieving state-of-the-art data efficiency.

Abstract

Solving partial differential equations with neural operators significantly reduces computational costs but remains bottlenecked by high training data requirements. Active learning offers a natural framework to mitigate this by selectively acquiring the most informative samples in an iterative manner. We introduce physics-based acquisition - a novel physics-informed active learning algorithm that leverages the partial differential equation residual to guide data selection. We validate the method by presenting numerical experiments for the 1D Burgers equation and the 2D compressible Navier-Stokes equations. We show that, in our experiments, physics-based acquisition consistently outperforms random acquisition and matches the state of the art in data efficiency. At the same time, it has the unique advantage of injecting a physics inductive bias into the training process, ensuring that simulation cost is spent where the model's physical understanding is weakest.

Problem

• neural operators
• partial differential equations
• data efficiency
• active learning
• physics-informed

Innovation

• neural operators
• active learning
• physics-informed
• PDE residual
• data efficiency