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Simultaneous parameter and function discovery in differential equations using physics-informed neural networks
Shalev Manor1, Mohammad Kohandel1
1Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada.
Mathematical Biosciences
|June 15, 2026
Summary
This study introduces a new framework for solving inverse problems in differential equations, ensuring unique solutions when identifying both parameters and functions simultaneously. This advances machine learning for complex scientific modeling.
Area of Science:
- Computational Science
- Applied Mathematics
- Scientific Machine Learning
Background:
- Inverse problems in differential equations are crucial for parameter and function identification from data.
- Existing methods like Physics-Informed Neural Networks (PINNs) and Universal Differential Equations (UDEs) struggle with simultaneous parameter and function identification due to non-unique solutions.
Purpose of the Study:
- To develop a novel framework guaranteeing unique solutions for inverse problems involving differential equations.
- To address the limitations of current machine learning approaches in simultaneously identifying unknown parameters and functions.
- To enhance the applicability of machine learning in modeling complex scientific and engineering systems.
Main Methods:
- Introduction of a new framework establishing conditions for guaranteed unique solutions in inverse problems.
- Application of the framework to biological and ecological dynamic systems.
- Utilizing machine learning techniques for parameter and function identification.
Main Results:
- Demonstrated accurate and interpretable results in biological and ecological modeling examples.
- Successfully addressed the challenge of solution non-uniqueness in simultaneous parameter and function identification.
- Established conditions for ensuring unique solutions in complex inverse problems.
Conclusions:
- The proposed framework significantly improves the ability to solve inverse problems involving differential equations.
- This work enhances the potential of machine learning techniques for modeling complex systems in science and engineering.
- The approach provides a pathway to more reliable and interpretable scientific modeling.
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