给物理神经网络加数学证明，让解的误差有严格保证。
Learn and Verify: A Framework for Rigorous Verification of Physics-Informed Neural Networks
- 用双光滑最大损失训练，提升解的稳定性
- 结合区间算术，给出可机器验证的误差界
- 适合需要高可信度科学计算的场景
利用神经网络求解微分方程已成为科学计算的核心课题，物理信息神经网络（PINNs）在正问题与逆问题中表现出强大能力。然而，与具有明确收敛保证的经典数值方法不同，基于神经网络的近似通常缺乏严格的误差界。此外，优化过程的非确定性使得其准确性难以数学认证。为此，我们提出“学习与验证”框架，为微分方程的解提供可计算、数学严格的误差界。通过结合新型双重光滑最大（DSM）损失进行训练，以及使用区间算术进行验证，我们计算出可被机器验证的后验误差界。在非线性常微分方程（ODEs）上的数值实验，包括含时变系数和有限时间爆破的问题，均表明该框架能成功构建真实解的严格包含区间，为可信科学机器学习奠定基础。
原文摘要 · Abstract (English)
The numerical solution of differential equations using neural networks has become a central topic in scientific computing, with Physics-Informed Neural Networks (PINNs) emerging as a powerful paradigm for both forward and inverse problems. However, unlike classical numerical methods that offer established convergence guarantees, neural network-based approximations typically lack rigorous error bounds. Furthermore, the non-deterministic nature of their optimization makes it difficult to mathematically certify their accuracy. To address these challenges, we propose a "Learn and Verify" framework that provides computable, mathematically rigorous error bounds for the solutions of differential equations. By combining a novel Doubly Smoothed Maximum (DSM) loss for training with interval arithmetic for verification, we compute rigorous a posteriori error bounds as machine-verifiable proofs. Numerical experiments on nonlinear Ordinary Differential Equations (ODEs), including problems with time-varying coefficients and finite-time blow-up, demonstrate that the proposed framework successfully constructs rigorous enclosures of the true solutions, establishing a foundation for trustworthy scientific machine learning.
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