Unsupervised Physics-Informed Operator Learning through Multi-Stage Curriculum Training
This addresses the problem of unstable convergence and limited generalization in physics-informed neural networks for researchers in scientific computing, representing an incremental improvement by combining existing techniques with a novel training strategy.
The paper tackles the challenge of solving partial differential equations in scientific machine learning by introducing a multi-stage physics-informed training strategy that achieves convergence comparable to supervised learning, using labeled data only along boundaries, with accuracy levels matching supervised methods across benchmarks.
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.