• Organisers: Marcelo Forets (@mforets) and Christian Schilling (@schillic)

• Moderator: David P. Sanders (@dpsanders)

A new generation of algorithms is addressing the fundamental challenge of how to exhaustively explore all possible scenarios for simulation of dynamical systems under model uncertainties. Moreover, deep neural networks play an increasing role in control and safety-critical applications, although it is often not known how to guarantee that they will behave correctly and safely under all circumstances. This minisymposium will host applications of set-based techniques and global optimization in Julia that address these questions.

We have made sure to reach out to several different groups who are working on set propagation techniques and their application in a wide range of areas, to present a broad overview of the area.

Plan: 1 introductory talk (non-Julia-specific), 5 regular talks, 1 Q&A panel.

• Introduction by Goran Frehse (Hybrid Systems Semantics group, Computer Science and System Engineering Laboratory (U2IS), ENSTA Paris. Homepage.

• Using Set Propagation and the Finite Element Method For Time Integration in Transient Solid Mechanics Problems. By Jorge Pérez Zerpa (speaker), Marcelo Forets and Daniel Freire Caporale. The Finite Element Method (FEM) is the gold standard for numerical simulation in transient solid mechanics problems. Several time-integration algorithms have been developed in recent decades; however, it is still a challenging problem to completely describe the family of dynamically-feasible behaviors from given sets of initial states. In this talk we take a set-based approach and conclude that it has a lot of potential to efficiently solve such problems.

• Dionysos.jl: Optimal Control of Cyber-Physical Systems. By Benoit Legat, Guillaume Berger, Julien Calbert (speaker) and Raphaël Jungers. Dionysos.jl is software produced by the ERC project Learning to Control (L2C). In view of the Cyber-Physical Systems (CPS) revolution, the only sensible way of controlling these complex systems is by discretizing the different variables, thus transforming the model into a simple combinatorial problem on a finite-state automaton, called an abstraction of this system. Our goal is to transform this approach into an effective, scalable, cutting-edge technology that will address the challenges of CPS and unlock their potential.

• Solving Optimization Problems with Embedded Dynamical Systems. By Matthew Wilhelm (speaker) and Matthew Stuber. We will discuss our recent work at PSORLab: EAGODynamicOptimizer.jl and DynamicBounds.jl packages. These extend our EAGO.jl nonconvex optimizer to address formulations containing embedded dynamical systems. We highlight a series of approaches for constructing the requisite convex and concave relaxations of differential equations in the original decision space and discuss the use of such techniques in a global optimization context. These methods may readily be composed with existing McCormick relaxation approaches, which allows for the solution of general nonlinear formulations to certified global optimality. Use cases relevant to hybrid data-driven process modeling, parameter estimation, and worst-case robust design are discussed.

• Computing with sets of probabilities in Julia. By Ander Gray. There are many ways to mathematically define a set of probability distributions, including: intervals, possibility distributions, random sets and probability boxes (p-boxes). These structures were discovered independently from one another, but are often synonymous and can be translated. Imprecise Probability theory links all these theories into one. In this presentation, we present ProbabilityBoundsAnalysis.jl (PBA) a numerical implementation of p-box arithmetic in Julia, which gives an arithmetic of random variables where both marginal distributions and dependencies may be partially defined. We show how PBA may be used to rigorously propagate distributions and p-boxes in reachability problems using ReachabilityAnalysis.jl.

• Methods to Soundly Verify Deep Neural Networks. By Tomer Arnon. Deep neural networks are widely used for nonlinear function approximation, with applications ranging from computer vision to control. Although these networks involve the composition of simple arithmetic operations, it can be very challenging to verify whether a particular network satisfies certain input-output properties. NeuralVerification.jl implements several methods that have emerged recently for soundly verifying such properties. We discuss fundamental differences between existing algorithms and compare them on a set of benchmark problems.