Solving time-dependent Partial Differential Equations (PDEs) is one of the most critical problems in computational science. While Physics-Informed Neural Networks (PINNs) offer a promising framework for approximating PDE solutions, their accuracy and training speed are limited by two core barriers: gradient-descent-based iterative optimization over complex loss landscapes and non-causal treatment of time as an extra spatial dimension. We present Frozen-PINN, a novel PINN based on the principle of space-time separation that leverages random features instead of training with gradient descent, and incorporates temporal causality by construction. On eight PDE benchmarks, including challenges such as extreme advection speeds, shocks, and high dimensionality, Frozen-PINNs achieve superior training efficiency and accuracy over state-of-the-art PINNs, often by several orders of magnitude. Our work addresses longstanding training and accuracy bottlenecks of PINNs, delivering quickly trainable, highly accurate, and inherently causal PDE solvers, a combination that prior methods could not realize. Our approach challenges the reliance of PINNs on stochastic gradient-descent-based methods and specialized hardware, leading to a paradigm shift in PINN training and providing a challenging benchmark for the community.

翻译：求解时间相关偏微分方程是计算科学中最关键的问题之一。尽管物理信息神经网络（PINNs）为近似偏微分方程解提供了有前景的框架，但其精度和训练速度受到两大核心障碍的限制：基于梯度下降的迭代优化处理复杂损失景观，以及将时间视为额外空间维度的非因果处理。我们提出Frozen-PINN，一种基于时空分离原理的新型PINN，它利用随机特征替代梯度下降训练，并通过构造方式融入时间因果性。在八个偏微分方程基准测试中，包括极端对流速度、激波和高维挑战，Frozen-PINNs在训练效率和精度上均优于最先进的PINNs，通常提升数个数量级。我们的工作解决了PINNs长期存在的训练与精度瓶颈，提供了可快速训练、高精度且内在因果的偏微分方程求解器，这是先前方法无法同时实现的。我们的方法挑战了PINNs对基于随机梯度下降方法和专用硬件的依赖，推动了PINN训练范式转变，并为学界提供了具有挑战性的基准。