Data-Driven Deep Learning of Partial Differential Equations in Modal Space
This work addresses the challenge of PDE discovery for researchers in computational science and engineering, though it is incremental as it builds on existing deep learning and modal space techniques.
The authors tackled the problem of recovering unknown time-dependent partial differential equations (PDEs) from solution data by approximating the evolution operator in a modal space using a deep neural network based on ResNet, achieving demonstrated effectiveness across various PDE types, including those with discontinuities like inviscid Burgers' equation.
We present a framework for recovering/approximating unknown time-dependent partial differential equation (PDE) using its solution data. Instead of identifying the terms in the underlying PDE, we seek to approximate the evolution operator of the underlying PDE numerically. The evolution operator of the PDE, defined in infinite-dimensional space, maps the solution from a current time to a future time and completely characterizes the solution evolution of the underlying unknown PDE. Our recovery strategy relies on approximation of the evolution operator in a properly defined modal space, i.e., generalized Fourier space, in order to reduce the problem to finite dimensions. The finite dimensional approximation is then accomplished by training a deep neural network structure, which is based on residual network (ResNet), using the given data. Error analysis is provided to illustrate the predictive accuracy of the proposed method. A set of examples of different types of PDEs, including inviscid Burgers' equation that develops discontinuity in its solution, are presented to demonstrate the effectiveness of the proposed method.