Table of contents : Preface Contents Contributors A Tree Structure Approach to Reachability Analysis 1 Introduction 2 Problem Setup 3 Backwards Reachability 4 Forwards Reachability 5 A Tree-Based Algorithm for Computing Reachable Sets 6 Numerical Examples 6.1 Numerical Example 1: Linear System with Two States 6.2 Numerical Example 2: DC Motor 7 Conclusions and Future Works References Asymptotic-Preserving Neural Networks for Hyperbolic Systems with Diffusive Scaling 1 Introduction 2 Hyperbolic Systems with Diffusive Scaling 3 Review of Deep Neural Networks and Physics-Informed Neural Networks 3.1 Deep Neural Networks (DNNs) 3.2 Physics-Informed Neural Networks (PINNs) 4 Asymptotic-Preserving Neural Networks 4.1 APNN for the Goldstein-Taylor Model 5 Application to the Goldstein-Taylor Model 5.1 Standard DNN Versus Standard PINN in Hyperbolic Regime 5.2 Standard PINN Versus APNN in Diffusive Regime 6 Application to Epidemic Dynamics 6.1 The Multiscale Hyperbolic SIR Model 6.2 APNN for the Hyperbolic SIR Model 6.3 APNN Performance with Epidemic Dynamics 7 Conclusion References A Non-local System Modeling Bi-directional Traffic Flows 1 Introduction 2 A Non-local Bi-directional Traffic Flow Model 3 Existence of Weak Solutions 4 Numerical Tests 4.1 Kernel Support Tending to Zero 4.2 Asymptotic Behaviour in a Periodic Setting 4.3 Maximum Principle References Semi-implicit Finite-Difference Methods for Compressible Gas Dynamics with Curved Boundaries: A Ghost-Point Approach 1 Introduction 2 Finite-Difference Methods for Conservation Laws 2.1 Compressible Euler Equations of Gas Dynamics 2.2 Explicit and Semi-implicit Spatial Discretization 2.3 Explicit and Semi-implicit Time Discretization 2.4 Discretization of Compressible Euler Equations in 2D 3 Ghost-Point Method for Boundary Conditions 3.1 Ghost-Point Technique for Implicit Solvers 4 Numerical Simulations 4.1 Square Obstacle 4.2 Circular Obstacle 4.3 Shock Tube Problems 5 Conclusions References High-Order Arbitrary-Lagrangian-Eulerian Schemes on Crazy Moving Voronoi Meshes 1 Introduction 1.1 Goals 1.2 Structure 2 Hyperbolic Partial Differential Equations 3 Numerical Method 3.1 Direct Arbitrary-Lagrangian-Eulerian Schemes 3.2 Topology Changes and Crazy Sliver Elements 3.3 ADER-ALE Algorithm: The Predictor Step 3.4 A Posteriori Sub-cell FV Limiter 4 Numerical Examples 4.1 Long Time Evolution of a Shu-type Vortical Equilibrium 4.2 Sedov Explosion Problem 4.3 Traveling Sod-type Explosion Problem 5 Conclusion and Outlook References Overview on Uncertainty Quantification in Traffic Models via Intrusive Method 1 Introduction 2 Stochastic Galerkin Approach 3 Microscopic Scale 4 Mesoscopic Scale 5 Macroscopic Scale 5.1 From Micro to Macro 5.2 From Meso to Macro 5.3 Numerical Test 6 Conclusion and Future Perspectives References A Study of Multiscale Kinetic Models with Uncertainties 1 Introduction 2 Mathematical Theory for Uncertain Kinetic Equations 2.1 Theoretical Framework: The Perturbative Setting 2.2 Convergence to the Global Equilibrium 3 Stochastic Galerkin Method: An Intrusive Scheme 3.1 Error Analysis of the gPC-SG System 3.2 Stochastic AP Schemes 4 Multi-fidelity Method: A Non-intrusive Scheme 4.1 A Bi-fidelity Stochastic Collocation (BFSC) Algorithm 4.2 Numerical Examples 5 Conclusion References On the Shock Wave Discontinuities in Grad Hierarchy for a Binary Mixture of Inert Gases 1 Introduction 2 13–Moment Equations and Principal Subsystems 3 The Shock Wave Problem 3.1 Singularity Manifolds and Critical Mach Numbers 4 Singularity Analysis References A Conservative a-Posteriori Time-Limiting Procedure in Quinpi Schemes 1 Introduction 2 Quinpi Scheme for Hyperbolic Conservation Laws 2.1 The Quinpi Approach 3 Numerical Tests 3.1 Test 1: Experimental Order of Convergence 3.2 Test 2: Linear Transport Problem 3.3 Test 3: Burgers Equation 3.4 Computational Performance of the Quinpi Schemes 4 Conclusion and Perspectives References Applications of Fokker Planck Equations in Machine Learning Algorithms 1 Introduction 2 Algorithmic Fairness for Imbalanced Data 2.1 Setting 2.2 Theorems 3 Asynchronous Stochastic Gradient Descent 3.1 Setting 3.2 Theorems 4 Reinforcement Learning in Smooth Environment 4.1 Setting 4.2 Theorems 5 Conclusion References