通过分阶段训练让神经算子在无监督下稳定求解偏微分方程。
Unsupervised Physics-Informed Operator Learning through Multi-Stage Curriculum Training
- 分阶段施加边界和内部残差，逐步优化损失函数
- 仅用边界数据即达到与有监督相当的精度
- 适合需要高稳定性与泛化能力的科学计算场景
求解偏微分方程仍是科学机器学习的核心挑战。神经算子通过学习函数空间间的映射，实现分辨率无关的推理，但通常依赖监督数据。物理信息神经网络虽可无监督训练，却常面临收敛不稳定和泛化能力差的问题。为此，我们提出多阶段物理信息训练策略：先逐步施加边界条件，再引入内部残差，并在每阶段重置优化器，起到延续机制作用，恢复稳定性并避免梯度停滞。我们进一步提出物理信息样条傅里叶神经算子（PhIS-FNO），结合傅里叶层与埃尔米特样条核，实现平滑残差评估。在典型基准上，PhIS-FNO仅使用窄边界标注数据，便达到与有监督学习相当的精度，确立了基于样条的分阶段优化在物理信息算子学习中的鲁棒性范式。
原文摘要 · Abstract (English)
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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