Providing non-trivial certificates of safety for non-linear stochastic systems is an important open problem. One promising solution to address this problem is the use of barrier functions. Barrier functions are functions whose composition with the system forms a Martingale and enable the computation of the probability that the system stays within a safe set over a finite time horizon. However, existing approaches to find barrier functions generally restrict the search to a small class of functions, often leading to conservatism. To address this problem, in this letter, we parameterize barrier functions as neural networks and show that bound propagation techniques and linear programming can be successfully employed to find Neural Barrier Functions. Further, we develop a branch-and-bound scheme based on linear relaxations that improves the scalability of the proposed framework. On several case studies we show that our approach scales to neural networks of hundreds of neurons and multiple hidden layers and often produces certificates of safety that are tighter than state-of-the-art methods.

Principles of feedback control have been shown to naturally arise in biological systems and have been applied with success to build synthetic circuits. Here, we present an implementation of a proportional–integral–derivative (PID) controller as a chemical reaction network with mass-action kinetics. This makes the controller synthesizable in vitro using DNA strand displacement technology, owing to its demonstrated capability of realizing arbitrary reaction-network designs as interacting DNA molecules. Previous related work has studied biological PID architectures using linearizations of nonlinear dynamics arising in both the controller components and in the plant. In this article, we present a proof of correctness of our nonlinear design in closed loop using arguments from singular perturbation theory. As an application to show the effectiveness of our controller, we provide numerical simulations on a genetic model to perform PID feedback control of protein expression.

Stochastic evolution of Chemical Reactions Networks (CRNs) over time is usually analyzed through solving the Chemical Master Equation (CME) or performing extensive simulations. Analysing stochasticity is often needed, particularly when some molecules occur in low numbers. Unfortunately, both approaches become infeasible if the system is complex and/or it cannot be ensured that initial populations are small. We develop a probabilistic logic for CRNs that enables stochastic analysis of the evolution of populations of molecular species. We present an approximate model checking algorithm based on the Linear Noise Approximation (LNA) of the CME, whose computational complexity is independent of the population size of each species and polynomial in the number of different species. The algorithm requires the solution of first order polynomial differential equations. We prove that our approach is valid for any CRN close enough to the thermodynamical limit. However, we show on four case studies that it can still provide good approximation even for low molecule counts. Our approach enables rigorous analysis of CRNs that are not analyzable by solving the CME, but are far from the deterministic limit. Moreover, it can be used for a fast approximate stochastic characterization of a CRN.

This work introduces a theoretical framework and a scalable computational method for formal analysis and control synthesis for switched diffusions, a class of stochastic models with linear dynamics that are continuous in both time and space domains; the focus is on safety with possible extensions to other properties. The proposed framework first constructs a finite abstraction in the form of an uncertain Markov process through discretization of both time and space domains. The errors caused by the discretization in each domain are formally characterized and cast into the abstraction model. Then, a strategy that maximizes the probability of the safety property and is robust against the errors is synthesized over the abstraction model. Finally, this robust strategy is mapped to a switching strategy for the stochastic processes that guarantees the safety property. The efficacy of the framework is demonstrated in three case studies, including one that illustrates the trade-off of the error contribution by the time and space discretization parameters.

Neural networks (NNs) are emerging as powerful tools to represent the dynamics of control systems with complicated physics or black-box components. Due to complexity of NNs, however, existing methods are unable to synthesize complex behaviors with guarantees for NN dynamic models (NNDMs). This letter introduces a control synthesis framework for stochastic NNDMs with performance guarantees. The focus is on specifications expressed in linear temporal logic interpreted over finite traces (LTLf), and the approach is based on finite abstraction. Specifically, we leverage recent techniques for convex relaxation of NNs to formally abstract a NNDM into an interval Markov decision process (IMDP). Then, a strategy that maximizes the probability of satisfying a given specification is synthesized over the IMDP and mapped back to the underlying NNDM. We show that the process of abstracting NNDMs to IMDPs reduces to a set of convex optimization problems, hence guaranteeing efficiency. We also present an adaptive refinement procedure that makes the framework scalable. On several case studies, we illustrate that our framework is able to provide non-trivial guarantees of correctness for NNDMs with architectures of up to 5 hidden layers and hundreds of neurons per layer.