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As a popular and easy-to-implement machine learning method for solvingdifferential equations, the physics-informed neural network (PINN) sometimesmay fail and find poor solutions which bias against the exact ones. In thispaper, we establish a framework of modified equation to explain the failurephenomenon and characterize the implicit bias of a general residualminimization (RM) method. We provide a simple way to derive the modifiedequation which models the numerical solution obtained by RM methods. Next, weshow the modified solution deviates from the original exact solution. The proofuses a by-product of this paper, that is, a necessary and sufficient conditionon characterizing the singularity of the coefficients. This equivalentcondition can be extended to other types of equations in the future. Finally,we prove, as a complete characterization of the implicit bias, that RM methodimplicitly biases the numerical solution against the exact solution and towardsa modified solution. In this work, we focus on elliptic equations withdiscontinuous coefficients, but our approach can be extended to other types ofequations and our understanding of the implicit bias may shed light on furtherdevelopment of deep learning based methods for solving equations.

On Residual Minimization for PDEs: Failure of PINN, Modified Equation, and Implicit Bias