A Unified Framework for Simultaneous Parameter and Function Discovery in Differential Equations
This addresses a bottleneck in modeling complex systems for scientists and engineers, though it appears incremental as it builds on prior methods like PINNs and UDEs.
The paper tackles the problem of simultaneously identifying unknown parameters and functions in differential equations from data, where existing methods like PINNs and UDEs face challenges with solution non-uniqueness. It introduces a framework that establishes conditions for unique solutions, demonstrating accurate results in biological and ecological systems.
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.