(46a) Dynamical-Systems-Guided Learning of PDEs from Data

Traditionally, a Partial Differential Equation (PDE) is derived by applying basic laws of physics (e.g. Newton's second law, conservation of energy) to one or several continuum variables. However, for many systems of current interest, like fluid systems sampled sparsely in space, we do not know physical laws that lead to analytical PDE forms. Yet, it is suspected that there may exist PDEs that could (approximately) predict the evolution of target variables in those systems. Recent data-driven methods have shown great promise for discovering such PDEs from experimental and computational data. These methods have potential drawbacks, such as dependence on training data, lack of guarantees for stability, and high computational cost compared to classical computational schemes. In this talk, we discuss some concepts from dynamical systems theory that underpin such data-driven approaches; through those concepts, we connect data-driven methods to a special class of physics-informed discretization schemes that can eliminate some of those drawbacks.

First, we investigate the problems where the PDE trajectories are attracted to a global low-dimensional manifold in the state space. For these problems, if the dimension of the underlying manifold is $n$ then knowledge of the solution at $2n+1$ spatial locations is enough to predict the evolution everywhere. This observation — justified by the well-known Whitney embedding theorem— allows us to learn multiple forms of PDEs that explain the evolution of the system. We demonstrate the application of this idea by learning a few PDEs for a reaction-diffusion system from data.

It is known that PDEs with strongly nonlinear behavior (e.g. turbulent fluid flows) do not possess a global low-dimensional attracting manifold. However, recent works on data-driven learning of coarse-grained PDEs suggest that post-transient low-dimensional behavior may still prevail locally. In particular, the neural-net-based architecture by Bar-Sinai et al. [1] learns the evolution of the solution profile restricted to a small spatial subdomain by using a smaller number of local measurements compared to, say, classical finite volume schemes. We justify this observation by appealing to a dynamical-systems coarse-graining of nonlinear PDEs known as “holistic discretization” [2]. This methodology, based on center manifold theory, gives a systematic approximation of the manifold that attracts the local solution profiles and hence provides relatively sparse but accurate discretizations for time stepping of the solution. Previous work, e.g. [3], has shown the effectiveness of this approach in guiding equation-free modeling of multi-scale systems.

Through examples from a linear advection-diffusion PDE and the nonlinear Burgers equation, we show that indeed the low-dimensional manifold exploited in data-driven methods coincides with the manifold approximated by the holistic discretization. We compare the performance of the holistic method to recent neural-net architectures, and explore the behavior of both approaches for a wide range of system parameters (Peclet/Reynolds number) and coarse-graining factors. An important feature of the holistic approach is that its stability and accuracy does not depend on the choice of the training data.

[1] Y. Bar-Sinai, S. Hoyer, J. Hickey, and M. P. Brenner, “Learning data-driven discretizations for partial differential equations,” Proceedings of the National Academy of Sciences 116, 15344–15349 (2019).

[2] A. Roberts, “Holistic discretization ensures fidelity to Burgers’ equation,” Applied numerical mathematics 37, 371–396 (2001).

[3] A. Roberts and I.G. Kevrekidis, ``General tooth boundary conditions for equation-free modeling", SIAM Journal of Scientific Computing 29.4, 1495-1510 (2007)