Whilst the partial differential equations that govern the dynamics of our world have been studied in great depth for centuries, solving them for complex, high-dimensional conditions and domains still presents an incredibly large mathematical and computational challenge. Analytical methods can be cumbersome to utilise, and numerical methods can lead to errors and inaccuracies. On top of this, sometimes we lack the information or knowledge to pose the problem well enough to apply these kinds of methods. Here, we present a new approach to approximating the solution to physical systems - physics-informed neural networks. The concept of artificial neural networks is introduced, the objective function is defined, and optimisation strategies are discussed. The partial differential equation is then included as a constraint in the loss function for the optimisation problem, giving the network access to knowledge of the dynamics of the physical system it is modelling. Some intuitive examples are displayed, and more complex applications are considered to showcase the power of physics informed neural networks, such as in seismic imaging. Solution error is analysed, and suggestions are made to improve convergence and/or solution precision. Problems and limitations are also touched upon in the conclusions, as well as some thoughts as to where physics informed neural networks are most useful, and where they could go next.

翻译：尽管支配我们世界动力学的偏微分方程已被深入研究多个世纪，但在复杂高维条件与领域中求解这些方程仍面临巨大的数学与计算挑战。解析方法应用繁琐，数值方法则可能产生误差与不精确性。此外，有时我们缺乏足够的信息或知识来充分定义问题从而应用这类方法。本文提出一种近似物理系统解的新方法——物理信息神经网络。我们介绍了人工神经网络的概念，定义了目标函数，并讨论了优化策略。随后将偏微分方程作为约束条件纳入优化问题的损失函数中，使网络能够获取其所建模物理系统的动力学知识。通过展示若干直观示例并考虑更复杂的应用（如地震成像），展现物理信息神经网络的强大能力。本文分析了求解误差，并提出改进收敛性和/或求解精度的建议。结论部分还涉及了问题与局限性，并探讨了物理信息神经网络最具应用前景的领域及其未来发展方向。