A method is presented that allows to reduce a problem described by differential equations with initial and boundary conditions to the problem described only by differential equations. The advantage of using the modified problem for physics-informed neural networks (PINNs) methodology is that it becomes possible to represent the loss function in the form of a single term associated with differential equations, thus eliminating the need to tune the scaling coefficients for the terms related to boundary and initial conditions. The weighted loss functions respecting causality were modified and new weighted loss functions based on generalized functions are derived. Numerical experiments have been carried out for a number of problems, demonstrating the accuracy of the proposed methods.

翻译：提出了一种将具有初始条件和边界条件的微分方程问题转化为仅由微分方程描述问题的方法。利用该修正问题处理物理信息神经网络（PINNs）方法的优势在于：可将损失函数表示为仅与微分方程相关的单项形式，从而无需调节边界条件和初始条件对应项的缩放系数。对遵循因果性的加权损失函数进行了修正，并推导出基于广义函数的新型加权损失函数。针对一系列问题开展了数值实验，验证了所提出方法的准确性。