Unsupervised Physics-Informed Operator Learning through Multi-Stage Curriculum Training

TL;DR

This paper introduces a multi-stage training approach for physics-informed neural operators that enhances convergence and generalization, using spline-enhanced Fourier neural operators to achieve supervised-level accuracy with minimal labeled data.

Contribution

It proposes a novel multi-stage curriculum training strategy and the PhIS-FNO model, combining spline kernels with Fourier layers for improved physics-informed operator learning.

Findings

Achieves supervised-level accuracy with limited boundary data.

Enhances training stability and convergence through staged optimization.

Demonstrates robustness across canonical benchmarks.

Abstract

Solving partial differential equations remains a central challenge in scientific machine learning. Neural operators offer a promising route by learning mappings between function spaces and enabling resolution-independent inference, yet they typically require supervised data. Physics-informed neural networks address this limitation through unsupervised training with physical constraints but often suffer from unstable convergence and limited generalization capability. To overcome these issues, we introduce a multi-stage physics-informed training strategy that achieves convergence by progressively enforcing boundary conditions in the loss landscape and subsequently incorporating interior residuals. At each stage the optimizer is re-initialized, acting as a continuation mechanism that restores stability and prevents gradient stagnation. We further propose the Physics-Informed Spline Fourier Neural Operator (PhIS-FNO), combining Fourier layers with Hermite spline kernels for smooth residual evaluation. Across canonical benchmarks, PhIS-FNO attains a level of accuracy comparable to that of supervised learning, using labeled information only along a narrow boundary region, establishing staged, spline-based optimization as a robust paradigm for physics-informed operator learning.