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.

翻译：在涉及偏微分方程弱形式算子与梯度流的数理建模中，损失函数的构建面临重大挑战，这主要源于需要恰当地选择测试函数。针对算子线性依赖于未知参数的情形，我们通过引入自测试损失函数来解决这一挑战，该函数采用依赖于未知参数的测试函数。所提出的自测试损失函数能够保持梯度流的能量守恒性质，并在随机微分方程情形下与期望对数似然比相一致。重要的是，该函数为二次型，便于对反问题的可辨识性与适定性进行理论分析，同时也能导出高效的参数化或非参数化回归算法。其计算过程简洁，仅需低阶导数甚至完全无需导数，数值实验表明该方法对含噪声及离散数据具有鲁棒性。