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

• URL: http://arxiv.org/abs/2505.16996v1
• Date: Thu, 22 May 2025 17:56:38 GMT
• Title: A Unified Framework for Simultaneous Parameter and Function Discovery in Differential Equations
• Authors: Shalev Manor, Mohammad Kohandel,
• Abstract summary: Inverse problems involving differential equations often require identifying unknown parameters or functions from data.<n>Existing approaches, such as Physics-Informed Neural Networks (PINNs), are effective at isolating either parameters or functions but can face challenges when applied simultaneously due to solution non-uniqueness.<n>We introduce a framework that addresses these limitations by establishing conditions under which unique solutions can be guaranteed.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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