Physics Informed Neurally Constructed ODE Networks (PINeCONes)

Maricela Best Mckay, Brian Wetton, Bhushan Gopaluni

Primary Area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)

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Keywords: Scientific Machine Learning, Neural ODEs, PINNs, PDEs

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Abstract: Recently, there has been a growing interest in using neural networks to approximate the solutions of partial differential equations (PDEs). Physics-informed neural networks (PINNs) have emerged as a promising framework for parameterizing PDE solutions using deep neural networks. However, PINNs often rely on memory-intensive optimizers to attain reasonable accuracy and can encounter training difficulties due to issues such as stiffness in the gradient flow of the loss. To address these challenges, we propose a novel network architecture that combines neural ordinary differential equations (ODEs) with physics-informed constraints in the loss function. In this approach, the dynamics within a neural ODE are expanded to include a system of ODEs whose solution provides the partial derivatives governing our PDE system. We call this architecture PINECONEs: physics-informed neurally constructed ODE networks. We evaluate the approach using simple but canonical PDEs from the literature to illustrate its potential. Our results show that training requires fewer iterations than previous approaches to achieve higher accuracy when using first-order optimization methods.

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Submission Number: 8482