Data-Driven Partial Derivative Equations Discovery with Evolutionary Approach

Abstract

The data-driven models allow one to dene the model structure in cases when a priori information is not sucient to build other types of models. The possible way to obtain physical interpretation is the data-driven differential equation discovery techniques. The existing methods of PDE (partial derivative equations) discovery are bound with the sparse regression. However, sparse regression is restricting the resulting model form, since the terms for PDE are defined before regression. The evolutionary approach described in the article has a symbolic regression as the background instead and thus has fewer restrictions on the PDE form. The evolutionary method of PDE discovery (EPDE) is described and tested on several canonical PDEs. The question of robustness is examined on a noised data example.

Symbolic regression in materials science

“…These equations are typically expressed in differential form, e.g., the Schrödinger equation (i d dt |Ψ(t) =Ĥ |Ψ(t) ) or Newton's second law (F = m dv dt ). It has been shown that symbolic regression can generate ordinary nonlinear partial differential equations for nonlinear coupled dynamical systems 46,68 as well as approximate ordinary differential equations. 69 Meanwhile, it is also often of desire to find conservation laws in physical systems.…”

Sparsistent Model Discovery

“…Furthermore, we restrict ourselves to cases where the original authors have tuned their algorithms and present the cases as being hard ones, see table 1. The results from the benchmark are presented in table 2 and figures 4 and 6 6 . Once the libraries are derived (with some noise), ∆(Θ, T ) 1, see table 1.…”

“…Usually F is identified based on an experiment consisting of n samples of the field u, see [2,3,4,5]. Some recent approaches use symbolic regression to find F, see [6], but so far the most popular approach to perform model discovery is by linear regression which was first introduced in [3] and consists in considering F as a linear combination of some candidate terms, u t = Θ • ξ, where each column in Θ is a candidate term for the underlying equation, typically a combination of polynomial and spatial derivative functions (e.g. u, u x , uu x ).…”

Towards Generative Design of Computationally Efficient Mathematical Models with Evolutionary Learning

“…As an example, the equation-free methods allow building the models that represent the multi-scale processes [ 12 ]. Another example is building of the physical laws from data in form of function [ 13 ], ordinary differential equations system [ 14 ], partial differential equations (PDE) [ 15 ]. The application of the automated design of ML models or pipelines (which are algorithmicaly close notions) are commonly named AutoML [ 8 ] although most of them work with models of fixed structure, some give opportunity to automatically construct relatively simple the ML structures.…”

“…A set of experiments have been held with the algorithm of data-driven partial differential equation discovery to analyze its performance with different task setups. All experiments were conducted using the EPDE framework described in detail in [ 15 ].…”