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A Unified Framework for Simultaneous Parameter and Function Discovery in Differential Equations

Shalev Manor, Mohammad Kohandel

TL;DR

This work addresses the challenge of non-uniqueness in inverse problems where both unknown constants and unknown functions are learned from differential-equation data. By deriving sufficiency conditions and proofs, the authors establish identifiability guarantees for combinations of constants and functions, and validate them across chemotherapy-intervention models and Lotka-Volterra dynamics. The framework enables simultaneous discovery of $\beta$, $\Psi$, and $u$, as well as unknown growth terms and functional components, using UPINNs and exact-derivative baselines. The results demonstrate accurate recovery under sufficient data coverage and robustness to noise, with clear guidelines on data requirements and limitations, suggesting broad applicability to scientific modeling problems with poorly understood dynamics.

Abstract

Inverse problems involving differential equations often require identifying unknown parameters or functions from data. Existing approaches, such as Physics-Informed Neural Networks (PINNs), Universal Differential Equations (UDEs) and Universal Physics-Informed Neural Networks (UPINNs), are effective at isolating either parameters or functions but can face challenges when applied simultaneously due to solution non-uniqueness. In this work, we introduce a framework that addresses these limitations by establishing conditions under which unique solutions can be guaranteed. To illustrate, we apply it to examples from biological systems and ecological dynamics, demonstrating accurate and interpretable results. Our approach significantly enhances the potential of machine learning techniques in modeling complex systems in science and engineering.

A Unified Framework for Simultaneous Parameter and Function Discovery in Differential Equations

TL;DR

This work addresses the challenge of non-uniqueness in inverse problems where both unknown constants and unknown functions are learned from differential-equation data. By deriving sufficiency conditions and proofs, the authors establish identifiability guarantees for combinations of constants and functions, and validate them across chemotherapy-intervention models and Lotka-Volterra dynamics. The framework enables simultaneous discovery of , , and , as well as unknown growth terms and functional components, using UPINNs and exact-derivative baselines. The results demonstrate accurate recovery under sufficient data coverage and robustness to noise, with clear guidelines on data requirements and limitations, suggesting broad applicability to scientific modeling problems with poorly understood dynamics.

Abstract

Inverse problems involving differential equations often require identifying unknown parameters or functions from data. Existing approaches, such as Physics-Informed Neural Networks (PINNs), Universal Differential Equations (UDEs) and Universal Physics-Informed Neural Networks (UPINNs), are effective at isolating either parameters or functions but can face challenges when applied simultaneously due to solution non-uniqueness. In this work, we introduce a framework that addresses these limitations by establishing conditions under which unique solutions can be guaranteed. To illustrate, we apply it to examples from biological systems and ecological dynamics, demonstrating accurate and interpretable results. Our approach significantly enhances the potential of machine learning techniques in modeling complex systems in science and engineering.

Paper Structure

This paper contains 22 sections, 8 theorems, 93 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Chemotherapy Intervention Example
  4. Problem of Uniqueness
  5. Establishing Uniqueness in Solutions
  6. Uniqueness with Known Temporal Scaling
  7. Generalization to Unknown Growth Terms
  8. Application of Uniqueness Conditions
  9. Case 1: Chemotherapy Intervention Problem
  10. Case 2: Modified Chemotherapy Intervention Problem
  11. Case 3: Lotka-Volterra Predator-Prey System
  12. Case 4: Lotka-Volterra Predator-Prey System with Proportional Noise
  13. Case 5: Chemotherapy Intervention Problem with Varying Dataset Length
  14. Results
  15. Chemotherapy Intervention Problem
  16. ...and 7 more sections

Key Result

Theorem 3.1

Suppose we have: where $\beta \in \mathbb{R}$ is an unknown constant, and $u: \mathbb{R}^k \to \mathbb{R}$ is an unknown function. The quantities $g(y), C(x), d(x)$, and the mapping $y = H_1(x)$ are all known. Suppose there exist two points $x_1, x_2 \in \mathbb{R}^n$ such that $y_1 = H_1(x_1) = H_1(x_2) = y_2$, $C(

Figures (3)

  • Figure 1: Top row: Solutions to the differential equations and the comparison between true and predicted drug action for Trial 1 ($u(N) = N$). Bottom row: Results for Trial 2 ($u(N) = N^2$).
  • Figure 2: Results for the modified chemotherapy intervention problem. Left: Solution to the differential equations. Right: Comparison of the true and predicted growth term $\Psi(N)$ and drug action $u(N)$.
  • Figure 3: Solution to the Lotka-Volterra system with the given initial conditions (left). Comparison between the true and predicted values of $\beta x$ and $\delta y$ (right).

Theorems & Definitions (17)

  • Theorem 3.1: Uniqueness with Known Temporal Scaling
  • proof
  • Corollary 3.2
  • Theorem 3.3: Uniqueness with Unknown Growth Terms
  • proof
  • Corollary 3.4
  • Theorem 3.5
  • Remark 3.6
  • Theorem 3.7
  • Remark 3.8
  • ...and 7 more