论文标题
使用牛顿的方法纠正神经网络近似来任意高阶隐式颂歌集成
Arbitrarily high order implicit ODE integration by correcting a neural network approximation with Newton's method
论文作者
论文摘要
作为通用近似的一种方法，深度神经网络（DNN）能够找到对问题的近似解决方案，而与适当的数学系统和目标函数相比，构成更大的限制。因此，与经典数值方法相比，DNN在应用中具有更大的灵活性。另一方面，他们对所追求的数学解决方案提供了不受控制的近似值。这表明将经典数值方法与基于DNN的近似值的杂交可能是理想的方法。在这项工作中，受到物理知识神经网络（PINN）方法启发的基于DNN的近似器，用于对牛顿的方法进行初步猜测，即对非线性ODES的高阶隐含runge-kutta（irk）集成，即洛伦兹系统。在通常的方法中，需要许多明确的时间段来为隐式系统的非线性求解器提供猜测，这需要足够的工作以使Irk方法不可行。这项工作中描述的基于DNN的方法使大型隐式时间阶段可以达到所需的准确性，以使DNN解决方案收集到几个百分之几的精度之内所需的努力。这项工作还开发了用于任意正交顺序的IRK方法矩阵元素的一般公式。
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.
