Data-Driven Theory-guided Learning of Partial Differential Equations using SimultaNeous Basis Function Approximation and Parameter Estimation (SNAPE)
This addresses the challenge of accurately modeling complex physical processes from highly noisy data, which is incremental as it builds on existing theory-guided learning approaches.
The authors tackled the problem of inferring governing partial differential equations (PDEs) from noisy spatiotemporal data by proposing SNAPE, a method that simultaneously fits basis functions and estimates parameters, achieving robustness against nearly 100% noise levels.
The measured spatiotemporal response of various physical processes is utilized to infer the governing partial differential equations (PDEs). We propose SimultaNeous Basis Function Approximation and Parameter Estimation (SNAPE), a technique of parameter estimation of PDEs that is robust against high levels of noise nearly 100 %, by simultaneously fitting basis functions to the measured response and estimating the parameters of both ordinary and partial differential equations. The domain knowledge of the general multidimensional process is used as a constraint in the formulation of the optimization framework. SNAPE not only demonstrates its applicability on various complex dynamic systems that encompass wide scientific domains including Schrödinger equation, chaotic duffing oscillator, and Navier-Stokes equation but also estimates an analytical approximation to the process response. The method systematically combines the knowledge of well-established scientific theories and the concepts of data science to infer the properties of the process from the observed data.