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| On Residual Minimization for PDEs: Failure ofPINN, Modified Equation, and Implicit Biase |
| Tao Luo,Qixuan Zhou |
| (School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai,
200240, China;
Institute of Natural Sciences, CMA-Shanghai, MOE-LSC and Qing Yuan Research Institute, Shanghai Jiao Tong University, Shanghai, 200240, China) |
| DOI： |
| Abstract: |
| 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. |
| Key words: residual minimization, PINN, implicit bias, modified equation. |