Abstract
Advances in machine learning (ML) coupled with increased computational power have enabled identification of patterns in data extracted from complex systems. ML algorithms are actively being sought in recovering physical models or mathematical equations from data. This is a highly valuable technique where models cannot be built using physical reasoning alone. In this paper, we investigate the application of fast function extraction (FFX), a fast, scalable, deterministic symbolic regression algorithm to recover partial differential equations (PDEs). FFX identifies active bases among a huge set of candidate basis functions and their corresponding coefficients from recorded snapshot data. This approach uses a sparsity-promoting technique from compressive sensing and sparse optimization called pathwise regularized learning to perform feature selection and parameter estimation. Furthermore, it recovers several models of varying complexity (number of basis terms). FFX finally filters out many identified models using non-dominated sorting and forms a Pareto front consisting of optimal models with respect to minimizing complexity and test accuracy. Numerical experiments are carried out to recover several ubiquitous PDEs such as wave and heat equations among linear PDEs and Burgers, Korteweg–de Vries (KdV), and Kawahara equations among higher-order nonlinear PDEs. Additional simulations are conducted on the same PDEs under noisy conditions to test the robustness of the proposed approach.
Highlights
Partial differential equations (PDEs) are ubiquitous in all branches of science and engineering.These mathematical models are generally derived from conservation laws, sound physical arguments, and empirical heuristics drawn from experiments by an insightful researcher
We demonstrate the use of fast function extraction (FFX), a deterministic symbolic regression algorithm, to identify and recover the target PDEs representing both linear and nonlinear dynamical systems
We look for the set of candidate solutions that are of minimum complexity and test error
Summary
Partial differential equations (PDEs) are ubiquitous in all branches of science and engineering. This framework faces challenge in recovering spatio-temporal data or high-dimensional measurements and highly correlated basis functions This limitation was addressed using sequential threshold ridge regression (STRidge) algorithm forming a framework called PDE functional identification of nonlinear dynamics (PDE-FIND) [20]. Elite base regression (EBR) [41] is a recent advancement in non-evolutionary computation where only elite bases are selected by measuring the correlation coefficient of basis functions with respect to the target model These elite bases are spanned in search space and use the parse matrix encoding scheme to propagate the algorithm further to recover mathematical model. We demonstrate the use of FFX, a deterministic symbolic regression algorithm, to identify and recover the target PDEs representing both linear and nonlinear dynamical systems.
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