H Schaeffer, Learning partial differential equations via data discovery and sparse optimization, Proc. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Williams, Gaussian processes for machine learning, Cambridge: MIT press, 2006. Karniadakis, Machine learning of linear differential equations using Gaussian processes, J. Karniadakis, Inferring solutions of differential equations using noisy multi-fidelity data, J. Owhadi, Bayesian numerical homogenization, Multiscale. Lipson, Distilling free-form natural laws from experimental data, Science, 324 (2009), 81–85. Lipson, Automated reverse engineering of nonlinear dynamical systems. This is achieved by relying on a relatively sparse amount of observation data obtained in combination with a selection of different initial data. Numerical experiments show that the proposed method has the ability to uncover the hidden conservation law for a wide variety of different nonlinear flux functions, ranging from pure concave/convex to highly non-convex shapes. We circumvent this obstacle by approximating the unknown conservation law (*) by an entropy satisfying discrete scheme where $ f(u) $ is represented through a symbolic multi-layer neural network. Secondly, the lack of regularity in the solution of (*) and the nonlinear form of $ f(u) $ hamper use of previous proposed physics informed neural network (PINN) methods where the underlying form of the sought differential equation is accounted for in the loss function. A main challenge with Eq (*) is that the solution typically creates shocks, i.e., one or several jumps of the form $ (u_L, u_R) $ with $ u_L \neq u_R $ moving in space and possibly changing over time such that information about $ f(u) $ in the interval associated with this jump is sparse or not at all present in the observation data. We propose a framework that combines a symbolic multi-layer neural network and a discrete scheme to learn the nonlinear, unknown flux function $ f(u) $ of the scalar conservation law A possible attractive approach is to extract conservation laws more directly from observation data by use of machine learning methods. However, deriving such nonlinear conservation laws is a significant and challenging problem.
Nonlinear conservation laws are widely used in fluid mechanics, biology, physics, and chemical engineering.
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