Physics-Informed Neural Networks (PINNs) have emerged as an influential technology, merging the swift and automated capabilities of machine learning with the precision and dependability of simulations grounded in theoretical physics. PINNs are often employed to solve algebraic or differential equations to replace some or even all steps of multi-stage computational workflows, leading to their significant speed-up. However, wide adoption of PINNs is still hindered by reliability issues, particularly at extreme ends of the input parameter ranges. In this study, we demonstrate this in the context of a system of coupled non-linear differential reaction-diffusion and heat transfer equations related to Fischer-Tropsch synthesis, which are solved by a finite-difference method with a PINN used in evaluating their source terms. It is shown that the testing strategies traditionally used to assess the accuracy of neural networks as function approximators can overlook the peculiarities which ultimately cause instabilities of the finite-difference solver. We propose a domain knowledge-based modifications to the PINN architecture ensuring its correct asymptotic behavior. When combined with an improved numerical scheme employed as an initial guess generator, the proposed modifications are shown to recover the overall stability of the simulations, while preserving the speed-up brought by PINN as the workflow component. We discuss the possible applications of the proposed hybrid transport equation solver in context of chemical reactors simulations.

翻译：物理信息神经网络（PINNs）已成为一项具有影响力的技术，它将机器学习的快速自动化能力与基于理论物理的模拟精度及可靠性相结合。PINNs常被用于求解代数或微分方程，以替代多阶段计算流程中的部分甚至全部步骤，从而实现显著的加速。然而，PINNs的广泛应用仍受可靠性问题的制约，特别是在输入参数范围的极端区域。本研究以费托合成相关的耦合非线性反应-扩散与传热微分方程组为例，通过有限差分法求解，并采用PINN评估其源项，从而揭示了这一问题。研究表明，传统用于评估神经网络作为函数逼近器准确性的测试策略可能会忽略某些特性，这些特性最终会导致有限差分求解器的不稳定性。我们提出了基于领域知识的PINN架构改进方案，确保其具有正确的渐近行为。当与作为初始猜测生成器的改进数值格式结合时，所提出的改进方案能够恢复模拟的整体稳定性，同时保持PINN作为工作流组件所带来的加速优势。我们探讨了所提出的混合输运方程求解器在化学反应器模拟中的潜在应用。