Learn and Verify: A Framework for Rigorous Verification of Physics-Informed Neural Networks
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A new "Learn and Verify" framework addresses the shortcomings of neural networks in solving differential equations by providing computable error bounds. It merges a Doubly Smoothed Maximum loss for training with interval arithmetic for verification, yielding rigorous a posteriori error bounds. Successful numerical tests on nonlinear ODEs show its potential for reliable scientific machine learning applications.
New Framework Addresses Accuracy Challenges in Physics-Informed Neural Networks
A novel "Learn and Verify" framework aims to enhance the reliability of Physics-Informed Neural Networks (PINNs) by providing computable, mathematically rigorous error bounds for solutions of differential equations. This addresses a key limitation of PINNs: the lack of rigorous error bounds and challenges in accuracy certification.
The framework integrates a Doubly Smoothed Maximum (DSM) loss function for training neural networks and employs interval arithmetic for verification, enabling the computation of rigorous a posteriori error bounds that can be mathematically validated.
Numerical Experiments Validate Effectiveness
Numerical experiments on nonlinear Ordinary Differential Equations (ODEs) demonstrate the framework's effectiveness, producing rigorous enclosures of true solutions even in challenging scenarios, such as:
- Problems with time-varying coefficients
- Situations involving finite-time blow-up
The results indicate a foundation for trustworthy applications of scientific machine learning.
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📰 Original Source: https://arxiv.org/abs/2601.19818v1
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