Physics-informed neural networks (PINNs) are an increasingly powerful way to
solve partial differential equations, generate digital twins, and create neural
surrogates of physical models. In this manuscript we detail the inner workings
of NeuralPDE.jl and show how a formulation structured around numerical
quadrature gives rise to new loss functions which allow for adaptivity towards
bounded error tolerances. We describe the various ways one can use the tool,
detailing mathematical techniques like using extended loss functions for
parameter estimation and operator discovery, to help potential users adopt
these PINN-based techniques into their workflow. We showcase how NeuralPDE uses
a purely symbolic formulation so that all of the underlying training code is
generated from an abstract formulation, and show how to make use of GPUs and
solve systems of PDEs. Afterwards we give a detailed performance analysis which
showcases the trade-off between training techniques on a large set of PDEs. We
end by focusing on a complex multiphysics example, the Doyle-Fuller-Newman
(DFN) Model, and showcase how this PDE can be formulated and solved with
NeuralPDE. Together this manuscript is meant to be a detailed and approachable
technical report to help potential users of the technique quickly get a sense
of the real-world performance trade-offs and use cases of the PINN techniques.

本文详细解释了物理启发神经网络（PINNs）内部运作机制，提出了结合数值积分的新的损失函数，介绍了扩展损失函数在参数估计和算子发现中的应用，并展示了如何使用纯符号公式生成全部的训练代码；随后给出了详细的性能分析来展示在大量偏微分方程（PDEs）上使用学习技术的权衡；最后，着重介绍了麻烦、复杂的多物理场例子，Doyle-Fuller-Newman（DFN）模型，展示了如何使用 NeuralPDE 将其表达并求解。这份论文旨在提供一个详细而易懂的技术报告，以帮助潜在的用户快速了解 PINN 技术的实际表现和使用案例。