Correctness Verification of Neural Networks Approximating Differential
  Equations
Petros Ellinas Rahul Nellikath Ignasi Ventura Jochen Stiasny Spyros Chatzivasileiadis
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.