As a popular and easy-to-implement machine learning method for solving differential equations, the physics-informed neural network (PINN) sometimes may fail and find poor solutions which bias against the exact ones. In this paper, we establish a framework of modified equation to explain the failure phenomenon and characterize the implicit bias of a general residual minimization (RM) method. We provide a simple way to derive the modified equation which models the numerical solution obtained by RM methods. Next, we show the modified solution deviates from the original exact solution. The proof uses a by-product of this paper, that is, a necessary and sufficient condition on characterizing the singularity of the coefficients. This equivalent condition can be extended to other types of equations in the future. Finally, we prove, as a complete characterization of the implicit bias, that RM method implicitly biases the numerical solution against the exact solution and towards a modified solution. In this work, we focus on elliptic equations with discontinuous coefficients, but our approach can be extended to other types of equations and our understanding of the implicit bias may shed light on further development of deep learning based methods for solving equations.

翻译：作为一种流行且易于实现的求解微分方程的机器学习方法，物理信息神经网络（PINN）有时可能会失效，并找到偏离精确解的低质量解。本文建立了修正方程的框架，以解释失效现象并刻画一般残差最小化（RM）方法的隐式偏差。我们提出了一种简单方法来推导描述RM方法数值解的修正方程。接着，我们证明修正解与原始精确解存在偏差。该证明利用了本文的一个副产品，即刻画系数奇异性的充要条件。这一等价条件未来可推广至其他类型的方程。最后，我们证明了RM方法隐式地将数值解偏向修正解而非精确解，从而完整刻画了其隐式偏差。本研究聚焦于具有间断系数的椭圆型方程，但所提出的方法可推广至其他方程类型，而对隐式偏差的理解可能为基于深度学习的方程求解方法提供进一步发展的启示。