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Towards true discovery of the differential equations

Alexander Hvatov, Roman Titov

TL;DR

This work addresses the challenge of discovering differential equations without expert input by moving beyond predefined term libraries. It proposes an expert-independent pipeline that combines token-based evolutionary discovery with versatile solvers and Bayesian-network–driven uncertainty assessment to map coefficient and solution uncertainties to data, rather than relying on a single fixed form. Through a Lynx-Hare (Lotka-Volterra) case study, the approach demonstrates the recovery of base dynamics and identification of higher-scale bases, with uncertainty quantified via joint distributions. The methodology enables robust, data-driven discovery of new governing relations, potentially extending to partially known or black-box systems while highlighting trade-offs in computational effort and automation needs.

Abstract

Differential equation discovery, a machine learning subfield, is used to develop interpretable models, particularly in nature-related applications. By expertly incorporating the general parametric form of the equation of motion and appropriate differential terms, algorithms can autonomously uncover equations from data. This paper explores the prerequisites and tools for independent equation discovery without expert input, eliminating the need for equation form assumptions. We focus on addressing the challenge of assessing the adequacy of discovered equations when the correct equation is unknown, with the aim of providing insights for reliable equation discovery without prior knowledge of the equation form.

Towards true discovery of the differential equations

TL;DR

This work addresses the challenge of discovering differential equations without expert input by moving beyond predefined term libraries. It proposes an expert-independent pipeline that combines token-based evolutionary discovery with versatile solvers and Bayesian-network–driven uncertainty assessment to map coefficient and solution uncertainties to data, rather than relying on a single fixed form. Through a Lynx-Hare (Lotka-Volterra) case study, the approach demonstrates the recovery of base dynamics and identification of higher-scale bases, with uncertainty quantified via joint distributions. The methodology enables robust, data-driven discovery of new governing relations, potentially extending to partially known or black-box systems while highlighting trade-offs in computational effort and automation needs.

Abstract

Differential equation discovery, a machine learning subfield, is used to develop interpretable models, particularly in nature-related applications. By expertly incorporating the general parametric form of the equation of motion and appropriate differential terms, algorithms can autonomously uncover equations from data. This paper explores the prerequisites and tools for independent equation discovery without expert input, eliminating the need for equation form assumptions. We focus on addressing the challenge of assessing the adequacy of discovered equations when the correct equation is unknown, with the aim of providing insights for reliable equation discovery without prior knowledge of the equation form.
Paper Structure (18 sections, 11 equations, 4 figures, 1 table)

This paper contains 18 sections, 11 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Current state of equation discovery
  3. Regression methods.
  4. Evolutionary optimization.
  5. Multi-objective step
  6. Answer 1: How to choose the discovery algorithm
  7. Uncertainty Estimation
  8. Answer 2: How to assess uncertainty
  9. Equation Solution
  10. Answer 3: How to solve discovered equations
  11. Equation discovery state-of-the-art
  12. Proposed approach for expert-independent equation discovery
  13. Experimental study
  14. Existing approach.
  15. Proposed approach.
  16. ...and 3 more sections

Figures (4)

  • Figure 1: Robust equation discovery pipeline: left - a group of tools used in the equation discovery, right - output of the tools with illustrations.
  • Figure 2: The results of solution sampled using Bayesian network systems for case (a) - first-order derivative restriction. Red lines - data, dashed line - 'mean' solution, corresponding intervals - maximum and minimum of an integrated function at a given time step.
  • Figure 3: The results of solution sampled using Bayesian network systems for case (b) - second-order derivative restriction. Red lines - data, dashed line - 'mean' solution, corresponding intervals - maximum and minimum of an integrated function at a given time step.
  • Figure 4: Example of Bayesian network for case (b). Index _u determines first equation, index _v determines the second.