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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.02264 |
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Table of Contents:
- 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.