(110i) Learning Partial Differential Equations from Discrete Space Time Data: Convolutional and Recurrent Networks, and Their Relations to Traditional Numerical Methods

In this contribution we develop and present extensions to previous methodologies [1] for the identification of distributed parameter systems from discrete space-time data based on artificial neural network architectures.

Earlier work relied on standard spatial numerical discretization techniques used for the solution of partial differential equations (PDEs), coupled to a multi-layer perceptron. Here, we demonstrate how these methods can be recast as a spatially convolutional neural network (CNN) integrated with a temporally recurrent network architecture based on numerical integrators [2,3,4], such as Runge-Kutta methods.

The relation between these architectures and "traditional" numerical algorithms is illustrated and discussed. We then focus on linking these tools with more elaborate nonlinear discretization schemes like the so-called "holistic discretization" of A. Roberts [5]. Finally, we explore porting these methods for learning "informed PDE discretization" to wider applications of of CNNs in machine learning, especially image analysis.

[1] R. González-García, R. Rico-Martínez, and I. G. Kevrekidis. "Identification of distributed parameter systems: a neural net based approach." Computers & Chemical Engineering, 1998.

[2] R. Rico-Martínez, K. Krischer, I. G. Kevrekidis, M. C. Kube, and J. L. Hudson. “Discrete- vs continuous-time nonlinear signal processing of Cu electrodissolution data.” Chemical Engineering Communications, 1992.

[3] R. Rico-Martínez, I. G. Kevrekidis, M. C. Kube, and J. L. Hudson. “Discrete- vs continuous-time nonlinear signal processing: Attractors, transitions and parallel implementation issues.” In American Control Conference, 1993.

[4] Ramiro Rico-Martínez and Ioannis G. Kevrekidis. “Nonlinear system identification using neural networks: dynamics and instabilities.” In Neural Networks for Chemical Engineers, edited by A. B. Bulsari, 409–42. Elsevier, 1995.

[5] Anthony J. Roberts. "A holistic finite difference approach models linear dynamics consistently." Mathematics of Computation, 2002.