Despite the success of physics-informed neural networks (PINNs) in approximating partial differential equations (PDEs), it is known that PINNs can sometimes fail to converge to the correct solution in problems involving complicated PDEs. This is reflected in several recent studies on characterizing and mitigating the ``failure modes'' of PINNs. While most of these studies have focused on balancing loss functions or adaptively tuning PDE coefficients, what is missing is a thorough understanding of the connection between failure modes of PINNs and sampling strategies used for training PINNs. In this paper, we provide a novel perspective of failure modes of PINNs by hypothesizing that the training of PINNs rely on successful ``propagation'' of solution from initial and/or boundary condition points to interior points. We show that PINNs with poor sampling strategies can get stuck at trivial solutions if there are propagation failures. We additionally demonstrate that propagation failures are characterized by highly imbalanced PDE residual fields where very high residuals are observed over very narrow regions. To mitigate propagation failures, we propose a novel evolutionary sampling (Evo) method that can incrementally accumulate collocation points in regions of high PDE residuals with little to no computational overhead. We provide an extension of Evo to respect the principle of causality while solving time-dependent PDEs. We theoretically analyze the behavior of Evo and empirically demonstrate its efficacy and efficiency in comparison with baselines on a variety of PDE problems.
翻译:尽管物理-知情神经网络(PINNs)在接近部分差异方程式(PDEs)方面取得成功,但众所周知,PINNs有时无法在涉及复杂PDEs的问题中找到正确的解决办法。这反映在最近关于PINNs“失灵模式”特征和减轻PINNs“失灵模式”特征的若干研究中。虽然这些研究大多侧重于平衡损失功能或适应性调PDE系数,但所缺少的是彻底了解PINNs失败模式与培训PINNs所使用的抽样战略之间的关联。在本文中,我们对PINNs失败模式的失败模式有了新的视角。我们通过虚报PINNs的培训依赖于成功地“调整”PINNs从初始和/或边界条件点到内部点的“失灵模式”的“失灵模式。虽然这些研究大多侧重于平衡损失功能或适应性调控PDE系数,但缺乏透彻的解决办法。我们进一步证明,传播失败的特征是高度不平衡的PDE残余领域,在非常狭窄的时间内观察到了非常高的残余现象。我们提议在不断递增的BIL(我们建议,在PDalalalalalalalalalation roduction rodustration rodustrislation) 中提供一种新的方法,但我们没有多少方法,我们没有提供一种在不断的递化方法,我们提供一种对PDocol-