提出自检验损失函数，解决微分方程建模中测试函数选择难题。
Self-test loss functions for learning weak-form operators and gradient flows
- 用依赖未知参数的测试函数构造损失，适用于线性算子情形。
- 损失函数为二次型，保证逆问题可识别且稳定，支持高效回归算法。
- 仅需低阶导数或无需导数，对噪声和离散数据鲁棒性强。
数据驱动建模中涉及偏微分方程弱形式算子与梯度流时，损失函数构造面临主要挑战，尤其在于测试函数的合理选取。本文提出自检验损失函数，其测试函数依赖于未知参数，专门针对算子关于未知量线性的情形。所提损失函数在梯度流中保持能量守恒，并与随机微分方程的期望对数似然比一致。重要的是，该损失为二次型，便于分析逆问题的可识别性与适定性，同时支持高效的参数化或非参数化回归算法。计算简单，仅需低阶导数甚至完全无导数，数值实验表明其对噪声和离散数据具有强鲁棒性。
原文摘要 · Abstract (English)
The construction of loss functions presents a major challenge in data-driven modeling involving weak-form operators in PDEs and gradient flows, particularly due to the need to select test functions appropriately. We address this challenge by introducing self-test loss functions, which employ test functions that depend on the unknown parameters, specifically for cases where the operator depends linearly on the unknowns. The proposed self-test loss function conserves energy for gradient flows and coincides with the expected log-likelihood ratio for stochastic differential equations. Importantly, it is quadratic, facilitating theoretical analysis of identifiability and well-posedness of the inverse problem, while also leading to efficient parametric or nonparametric regression algorithms. It is computationally simple, requiring only low-order derivatives or even being entirely derivative-free, and numerical experiments demonstrate its robustness against noisy and discrete data.
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