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

• URL: http://arxiv.org/abs/2602.02264v1
• Date: Mon, 02 Feb 2026 16:06:57 GMT
• Title: Unsupervised Physics-Informed Operator Learning through Multi-Stage Curriculum Training
• Authors: Paolo Marcandelli, Natansh Mathur, Stefano Markidis, Martina Siena, Stefano Mariani,
• Abstract summary: We introduce a physics-informed training strategy that achieves convergence by enforcing boundary conditions in the loss landscape.<n>At each stage the limitation is re-formed, acting as a continuation mechanism that restores stability and prevents stagnation.<n>Across canonical benchmarks, PhIS-FNO attains a level of accuracy comparable to that of supervised learning.
• Score: 1.5620806570871846
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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