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.

本文提出了一种新的物理信息神经网络故障模式的解释方式，并从采样策略的角度解释了其失败模式。此外，作者还提出了一种新的演化抽样方法，可以有效地缓解此类问题。

利用进化采样缓解 PINNs 中的传播故障

Mitigating Propagation Failures in PINNs using Evolutionary Sampling

Perturb and Combine (P&C) group of methods generate multiple versions of the
predictor by perturbing the training set or construction and then combining
them into a single predictor (Breiman, 1996b). The motive is to improve the
accuracy in unstable classification and regression methods. One of the most
well known method in this group is Bagging. Arcing or Adaptive Resampling and
Combining methods like AdaBoost are smarter variants of P&C methods. In this
extended abstract, we lay the groundwork for a new family of methods under the
P&C umbrella, known as Evolutionary Sampling (ES). We employ Evolutionary
algorithms to suggest smarter sampling in both the feature space (sub-spaces)
as well as training samples. We discuss multiple fitness functions to assess
ensembles and empirically compare our performance against randomized sampling
of training data and feature sub-spaces.

本研究介绍了 Perturb and Combine（P＆C）方法的一种新型演化采样法（ES），该方法采用进化算法以智能方式采样特征空间和训练样本，并使用多个适应性函数评估合奏效果，与随机采样的比较表明，此方法对于不稳定分类和回归方法的准确性有显著提高。