Incorporating Neural ODEs into DAE-Constrained Optimization Problems
Keywords: Differential Algebraic Equations, Neural Ordinary Differential Equations, Dynamic Optimization, Hybrid Modeling
TL;DR: This paper presents a novel approach to solving DAE-constrained optimization problems by incorporating Neural Ordinary Differential Equations, demonstrating enhanced performance through realistic case studies in various dynamic systems.
Abstract: Differential algebraic equations (DAEs) are pivotal in dynamic optimization across diverse fields, from process control to flight trajectory optimization and epidemiological modeling. Traditional methods like single shooting, multiple shooting, and direct transcription effectively optimize known mechanistic models. However, significant challenges arise when the underlying equations are unknown or deviate from empirical data. While black-box optimization strategies can address some issues, challenges persist regarding data quality, non-linearity, and the inclusion of constraints. Recent advances in machine learning, particularly Neural ODEs, offer promising tools for continuous representation of dynamic systems. This work bridges the gap between machine learning representations of dynamic systems and optimization methodologies, enabling a novel approach for solving DAEs with data-driven components. We demonstrate this approach using numerical examples of DAE problems and realistic case studies, including biochemical reactor control and disease spread prevention. Our results highlight the efficacy of incorporating Neural ODEs into equation-based solvers, showing improved performance over existing strategies such as SINDy. Additionally, we formalize the optimization program for NN-embedded DAEs and present representations for common neural network architectures (e.g., ReLU, tanh). This work contributes a novel framework for dynamic system optimization, integrating machine learning advancements with traditional optimization techniques, and offers practical insights through comprehensive case studies.
Primary Area: optimization
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Submission Number: 5234
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