Deep Neural Networks as Discrete Dynamical Systems: Implications for Physics-Informed Learning
arXiv:2601.00473v3 Announce Type: replace Abstract: We revisit the analogy between feed-forward deep neural networks (DNNs) and discrete dynamical systems derived from neural integral equations and their corresponding partial differential equation (PDE) forms. A comparative analysis between the numerical/exact solutions of the Burgers' and Eikonal equations, and the same obtained via PINNs is presented. We show that PINN learning provides a different computational pathway compared to standard numerical discretization in approximating essentially the same underlying dynamics of the system. With
This research is published as AI models become increasingly complex, requiring new methods for understanding their fundamental properties and potential applications, especially in scientific computing.
A strategic reader should care because understanding DNNs as discrete dynamical systems can lead to more robust, interpretable, and efficient AI, critical for scientific modeling and complex system simulations.
The perspective of DNNs as dynamical systems changes how their learning pathways are conceptualized, potentially bridging AI computation with traditional numerical methods for solving physical equations.
- · AI researchers
- · Scientific computing sector
- · Physics-informed AI developers
- · Engineering simulations
- · Traditional numerical methods (potentially less efficient)
- · Black-box AI approaches without theoretical grounding
Improved understanding and design principles for Physics-Informed Neural Networks (PINNs).
Accelerated development of AI for scientific discovery and engineering, enabling faster prototyping and simulation.
The integration of AI into complex system design becomes seamless, leading to breakthroughs in fields like materials science or climate modeling.
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