Self-Supervised Learning with Lie Symmetries for Partial Differential Equations

Dynamical systems governed by differential equations are ubiquitous in fluid dynamics, chemistry, astrophysics, and beyond. Accurately analyzing and predicting the evolution of such systems is of paramount importance, inspiring decades of innovation in algorithms for numerical methods. However, high-accuracy solvers are often computationally expensive. Machine learning has recently arisen as an alternative method for analyzing differential equations at a fraction of the cost [1, 2, 3]. Typically, the neural network for a given equation is trained on simulations of that same equation, generated by numerical solvers that are high-accuracy but comparatively slow [4]. What if we instead wish to learn from heterogeneous data, e.g., data with missing information, or gathered from actual observation of varied physical systems rather than clean simulations? For example, we may have access to a dataset of instances of time-evolution, stemming from a family of partial differential equations (PDEs) for which important characteristics of the problem, such as viscosity or initial conditions, vary and/or are unknown. In this case, representations learned from such a large, "unlabeled" dataset could still prove useful in learning to identify unknown characteristics, given only a small dataset "labeled" with viscosities or reaction constants. Alternatively, the "unlabeled" dataset may contain evolutions over very short periods of time, or with missing time intervals; possible goals are then to learn representations that could be useful in filling in these gaps, or regressing other quantities of interest.