# Correctness Verification of Neural Networks Approximating Differential Equations

arXiv: https://arxiv.org/abs/2402.07621 · HTML: https://mlsanity.com/paper/2402.07621/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 30 references that are in the map.
- First posted on arXiv 12 Feb 2024.

## Builds on (most cited first, 16 of 16)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models](https://mlsanity.com/paper/1810.12715/) (2018)
- [Fast and Complete: Enabling Complete Neural Network Verification with Rapid and Massively Parallel Incomplete Verifiers](https://mlsanity.com/paper/2011.13824/) (2020)
- [MNIST-C: A Robustness Benchmark for Computer Vision](https://mlsanity.com/paper/1906.02337/) (2019)
- [Robustness Verification for Transformers](https://mlsanity.com/paper/2002.06622/) (2020)
- [Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs](https://mlsanity.com/paper/2106.14473/) (2021)
- [Physics-Informed Neural Networks for AC Optimal Power Flow](https://mlsanity.com/paper/2110.02672/) (2021)
- [Physics-Informed Neural Networks for Time-Domain Simulations: Accuracy, Computational Cost, and Flexibility](https://mlsanity.com/paper/2303.08994/) (2023)
- [Stability Verification of Neural Network Controllers using Mixed-Integer Programming](https://mlsanity.com/paper/2206.13374/) (2022)
- [Correctness Verification of Neural Networks](https://mlsanity.com/paper/1906.01030/) (2019)
- [Neural Network Branch-and-Bound for Neural Network Verification](https://mlsanity.com/paper/2107.12855/) (2021)
- [Physics-informed neural networks for predicting gas flow dynamics and unknown parameters in diesel engines](https://mlsanity.com/paper/2304.13799/) (2023)
- [Minimizing Worst-Case Violations of Neural Networks](https://mlsanity.com/paper/2212.10930/) (2022)

## Built on by (most cited first, 1 of 1)

- [A Unified Approach for Learning the Dynamics of Power System Generators and Inverter-based Resources](https://mlsanity.com/paper/2409.14454/) (2024)

## Abstract (arXiv)

Verification of Neural Networks (NNs) that approximate the solution of Partial Differential Equations (PDEs) is a major milestone towards enhancing their trustworthiness and accelerating their deployment, especially for safety-critical systems. If successful, such NNs can become integral parts of simulation software tools which can accelerate the simulation of complex dynamic systems more than 100 times. However, the verification of these functions poses major challenges; it is not straightforward how to efficiently bound them or how to represent the derivative of the NN. This work addresses both these problems. First, we define the NN derivative as a finite difference approximation. Then, we formulate the PDE residual bounding problem alongside the Initial Value Problem's error propagation. Finally, for the first time, we tackle the problem of bounding an NN function without a priori knowledge of the output domain. For this, we build a parallel branching algorithm that combines the incomplete CROWN solver and Gradient Attack for termination and domain rejection conditions. We demonstrate the strengths and weaknesses of the proposed framework, and we suggest further work to enhance its efficiency.