Physics-informed neural networks (PINNs) have gained significant prominence as a powerful tool in the field of scientific computing and simulations. Their ability to seamlessly integrate physical principles into deep learning architectures has revolutionized the approaches to solving complex problems in physics and engineering. However, a persistent challenge faced by mainstream PINNs lies in their handling of discontinuous input data, leading to inaccuracies in predictions. This study addresses these challenges by incorporating the discretized forms of the governing equations into the PINN framework. We propose to combine the power of neural networks with the dynamics imposed by the discretized differential equations. By discretizing the governing equations, the PINN learns to account for the discontinuities and accurately capture the underlying relationships between inputs and outputs, improving the accuracy compared to traditional interpolation techniques. Moreover, by leveraging the power of neural networks, the computational cost associated with numerical simulations is substantially reduced. We evaluate our model on a large-scale dataset for the prediction of pressure and saturation fields demonstrating high accuracies compared to non-physically aware models.

翻译：物理信息神经网络（PINNs）作为科学计算与模拟领域的强大工具已获得显著关注。其将物理原理无缝融入深度学习架构的能力，为解决物理与工程中的复杂问题带来了范式革新。然而，主流PINNs在处理非连续输入数据时仍面临持续挑战，导致预测精度不足。本研究通过将控制方程的离散化形式纳入PINN框架来应对这些挑战。我们提出将神经网络的能力与离散微分方程所施加的动力学相结合。通过对控制方程进行离散化处理，PINN能够学习处理非连续性并准确捕捉输入与输出间的潜在关联，相较于传统插值技术显著提升了预测精度。此外，借助神经网络的计算优势，数值模拟相关的计算成本得到大幅降低。在用于预测压力场与饱和度场的大规模数据集上评估表明，与缺乏物理意识的模型相比，本方法展现出更高的预测精度。