Recent advances in deep learning have allowed neural networks (NNs) to successfully replace traditional numerical solvers in many applications, thus enabling impressive computing gains. One such application is time domain simulation, which is indispensable for the design, analysis and operation of many engineering systems. Simulating dynamical systems with implicit Newton-based solvers is a computationally heavy task, as it requires the solution of a parameterized system of differential and algebraic equations at each time step. A variety of NN-based methodologies have been shown to successfully approximate the trajectories computed by numerical solvers at a fraction of the time. However, few previous works have used NNs to model the numerical solver itself. For the express purpose of accelerating time domain simulation speeds, this paper proposes and explores two complementary alternatives for modeling numerical solvers. First, we use a NN to mimic the linear transformation provided by the inverse Jacobian in a single Newton step. Using this procedure, we evaluate and project the exact, physics-based residual error onto the NN mapping, thus leaving physics ``in the loop''. The resulting tool, termed the Physics-pRojected Neural-Newton Solver (PRoNNS), is able to achieve an extremely high degree of numerical accuracy at speeds which were observed to be up to 31% faster than a Newton-based solver. In the second approach, we model the Newton solver at the heart of an implicit Runge-Kutta integrator as a contracting map iteratively seeking a fixed point on a time domain trajectory. The associated recurrent NN simulation tool, termed the Contracting Neural-Newton Solver (CoNNS), is embedded with training constraints (via CVXPY Layers) which guarantee the mapping provided by the NN satisfies the Banach fixed-point theorem.
翻译:最近深层学习的进展使得神经网络(NNS)能够成功地取代许多应用中的传统数字求解器,从而获得令人印象深刻的计算收益。这样的应用之一是时间域模拟,这是许多工程系统的设计、分析和运行所不可或缺的。用隐含的牛顿型求解器模拟动态系统是一项计算上繁重的任务,因为它要求每一步都有一个差异和代数方程参数化系统的解决办法。基于NNW的各种方法已经显示能够成功地近似数字求解器在一定时间内计算出的传统数字求解器的轨迹。然而,以前没有多少作品使用NNS模拟数字求解器本身。为了加速时间域模拟速度,本文提议和探索两个模型数字求解答器的互补替代方案。首先,我们用NNE(NN)来模拟每一步由反向雅各布提供的差异和代数方方方方方程式提供的线性转换。我们用这个程序评估和预测NNF绘图中基于物理的精确、基于物理的余积积存错误,从而留下“物理学”的内核“内存”的内径解器。由此产生的工具在Sl-Sl-SlSeral Stal Streal StalS,可以实现一个高的内点的内置的内置的内值。