用牛顿法修正神经网络逼近的高阶隐式ODE积分
Arbitrarily high order implicit ODE integration by correcting a neural network approximation with Newton's method
摘要：
作为一种通用近似方法，深度神经网络（DNN）能够找到问题的近似解，而这些问题的约束条件仅比适当设定的数学系统和目标函数多一点。因此，DNN在应用中比经典的数值方法具有更大的灵活性。另一方面，它们提供了一个不受控制的近似追求的数学解决方案。这表明经典数值方法与基于DNN的近似的混合可能是一种理想的方法。在这项工作中，一个基于DNN的近似器的物理信息的神经网络（PINNs）方法的启发被用来提供一个初始猜测的牛顿法迭代的一个非常高阶的隐式龙格库塔（IRK）积分的非线性系统的常微分方程，即洛伦兹系统。在通常的方法中，需要许多显式时间步长来为隐式系统的非线性求解器提供猜测，这需要足够的工作来使IRK方法不可行。在这项工作中描述的基于DNN的方法使大隐式时间步长能够达到任何所需的准确度，以便尽可能多地将DNN解决方案收敛到百分之几的准确度。这项工作还开发了一个通用公式的矩阵元素的IRK方法的任意求积阶。
英文摘要：
As a method of universal approximation deep neural networks (DNNs) are capable of finding approximate solutions to problems posed with little more constraints than a suitably-posed mathematical system and an objective function. Consequently, DNNs have considerably more flexibility in applications than classical numerical methods. On the other hand they offer an uncontrolled approximation to the sought-after mathematical solution. This suggests that hybridization of classical numerical methods with DNN-based approximations may be a desirable approach. In this work a DNN-based approximator inspired by the physics-informed neural networks (PINNs) methodology is used to provide an initial guess to a Newton's method iteration of a very-high order implicit Runge-Kutta (IRK) integration of a nonlinear system of ODEs, namely the Lorenz system. In the usual approach many explicit timesteps are needed to provide a guess to the implicit system's nonlinear solver, requiring enough work to make the IRK method infeasible. The DNN-based approach described in this work enables large implicit time-steps to be taken to any desired degree of accuracy for as much effort as it takes to converge the DNN solution to within a few percent accuracy. This work also develops a general formula for the matrix elements of the IRK method for an arbitrary quadrature order.
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