Joshua Pilipovsky

Joshua Pilipovsky, Vignesh Sivaramakrishnan, Meeko M. K. Oishi et al.

Verifying the input-output relationships of a neural network so as to achieve some desired performance specification is a difficult, yet important, problem due to the growing ubiquity of neural nets in many engineering applications. We use ideas from probability theory in the frequency domain to provide probabilistic verification guarantees for ReLU neural networks. Specifically, we interpret a (deep) feedforward neural network as a discrete dynamical system over a finite horizon that shapes distributions of initial states, and use characteristic functions to propagate the distribution of the input data through the network. Using the inverse Fourier transform, we obtain the corresponding cumulative distribution function of the output set, which can be used to check if the network is performing as expected given any random point from the input set. The proposed approach does not require distributions to have well-defined moments or moment generating functions. We demonstrate our proposed approach on two examples, and compare its performance to related approaches.

Joshua Pilipovsky

In this paper we develop a sequential convex programming (SCP) framework for free-final-time covariance steering of nonlinear stochastic differential equations (SDEs) subject to both additive and multiplicative diffusion. We cast the free-final-time objective through a time-normalization and introduce per-interval time-dilation variables that induce an adaptive discretization mesh, enabling the simultaneous optimization of the control policy and the temporal grid. A central difficulty is that, under multiplicative noise, accurate covariance propagation within SCP requires retaining the first-order diffusion linearization and its coupling with time dilation. We therefore derive the exact local linear stochastic model (preserving the multiplicative structure) and introduce a tractable discretization that maintains the associated diffusion terms, after which each SCP subproblem is solved via conic/semidefinite covariance-steering relaxations with terminal moment constraints and state/control chance constraints. Numerical experiments on a nonlinear double-integrator with drag and velocity-dependent diffusion validate free-final-time minimization through adaptive time allocation and improved covariance accuracy relative to frozen-diffusion linearizations.

Vignesh Sivaramakrishnan, Krishna C. Kalagarla, Rosalyn Devonport et al.

We present a neural network verification toolbox to 1) assess the probability of satisfaction of a constraint, and 2) synthesize a set expansion factor to achieve the probability of satisfaction. Specifically, the tool box establishes with a user-specified level of confidence whether the output of the neural network for a given input distribution is likely to be contained within a given set. Should the tool determine that the given set cannot satisfy the likelihood constraint, the tool also implements an approach outlined in this paper to alter the constraint set to ensure that the user-defined satisfaction probability is achieved. The toolbox is comprised of sampling-based approaches which exploit the properties of signed distance function to define set containment.