A Pontryagin maximum principle on the belief space for continuous-time stochastic optimal control with discrete observations

We study a continuous time stochastic optimal control problem under partial observations that are available only at discrete time instants. This hybrid setting, with continuous dynamics and intermittent noisy measurements, arises in applications ranging from robotic exploration and target tracking to epidemic control. We formulate the problem on the space of beliefs (information states), treating the controller’s posterior distribution of the state as the state variable for decision making. On this belief space we derive a Pontryagin maximum principle that provides necessary conditions for optimality. The analysis carefully tracks both the continuous evolution of the state between observation times and the Bayesian jump updates of the belief at observation instants. A key insight is a relationship between the adjoint process in our maximum principle and the gradient of the value functional on the belief space, which links the optimality conditions to the dynamic programming approach on the space of probability measures. The resulting optimality system has a prediction and update structure that is closely related to the unnormalised Zakai equation and the normalised Kushner-Stratonovich equation in nonlinear filtering. Building on this analysis, we design a particle based numerical scheme to approximate the coupled forward (filter) and backward (adjoint) system. The scheme uses particle filtering to represent the evolving belief and regression techniques to approximate the adjoint, which yields a practical algorithm for computing near optimal controls under partial information. The effectiveness of the approach is illustrated on both linear and nonlinear examples and highlights in particular the benefits of actively controlling the observation process

Regularity for non-autonomous parabolic equations with right-hand side singular measures involved

This article provides a theory for non-autonomous parabolic equations the right hand side of which includes singular measures - depending on the time parameter - on the spatial domain. In two space dimensions all bounded Radon measures are admissable as such. In higher dimensions the focus is on measures whose support is concentrated on l-sets in the sense of Jonsson and Wallin. It is shown that they may interpreted as elements from a Sobolev space W. So the right hand side is considered as an element from a W-valued Lebesgue space on the time interval. Having this at hand, previous results on maximal (non-autonomous) maximal parabolic regularity apply and show that the solution lies in the corresponding space of maximal parabolic regularity. In contrast to other work in this field we only require absolute minimal smothness for the data of the problem: the domain, the coefficients - and mixed boundary conditions are allowed. Under minimally stronger assumptions we even show the Hölder property in space and time. Overall, this work contains an interplay of geometric measure theory with advanced parabolic theory which delivers as much parabolic regularity for the solution as one can maximally expect

Time-asymptotic self-similarity of the damped compressible Euler equations in parabolic scaling variables

We study the long-time behavior of solutions to the compressible Euler equations with frictional damping in the whole space, where we prescribe direction-dependent values for the density at spatial infinity. To this end, we transform the system into parabolic scaling variables and derive a relative entropy inequality, which allows to conclude the convergence of the density towards a self-similar solution to the porous medium equation while the associated limit momentum is governed by Darcy's law. Moreover, we obtain convergence rates that explicitly depend on the flatness of the limit profile. While we focus on weak solutions in the one-dimensional case, we extend our results to energy-variational solutions in the multi-dimensional setting

Layerwise goal-oriented adaptivity for neural ODEs: An optimal control perspective

In this work, we propose a novel layerwise adaptive construction method for neural network architectures. Our approach is based on a goal--oriented dual-weighted residual technique for the optimal control of neural differential equations. This leads to an ordinary differential equation constrained optimization problem with controls acting as coefficients and a specific loss function. We implement our approach on the basis of a DG(0) Galerkin discretization of the neural ODE, leading to an explicit Euler time marching scheme. For the optimization we use steepest descent. Finally, we apply our method to the construction of neural networks for the classification of data sets, where we present results for a selection of well known examples from the literature

Renormalised solutions to reaction-diffusion systems with interface conditions: Global existence and weak-strong uniqueness

We introduce an extension of the concept of renormalised solutions for entropy-dissipating reaction-diffusion systems due to J. Fischer (Arch. Ration. Mech. Anal. 218, 2015) to systems coupled by nonlinear interface conditions. For this notion of solution, we establish global existence as well as a weak-strong stability estimate. Our framework allows to handle entropy-dissipating interfacial transmission rates without growth restrictions, including power-law nonlinearities as arising in the thermodynamic modelling of dissipative bulk-interface systems via generalised gradient structures. Our analysis relies on suitable extensions of the species' densities across the interface as well as on a non-local truncated variant of the relative entropy

On the equilibrium solutions of electro-energy-reaction-diffusion systems

Electro-energy-reaction-diffusion systems are thermodynamically consistent continuum models for reaction-diffusion processes that account for temperature and electrostatic effects in a way that total charge and energy are conserved. The question of the long-time asymptotic behavior in electro-energy-reaction-diffusion systems and the characterization of their equilibrium solutions leads to a maximization problem of the entropy on the manifold of states with fixed values for the linear charge and the nonlinear convex energy functional. As the main result, we establish the existence, uniqueness, and regularity of solutions to this constrained optimization problem. We give two conceptually different proofs, which are related to different perspectives on the constrained maximization problem. The first one is based on the Lagrangian approach, whereas the second one employs the direct method of the calculus of variations

Gradient-robustness for the isentropic compressible Stokes problem

This paper extends a scheme for the isothermal compressible Stokes problem to the isentropic compressible Stokes problem that has two important ingredients. The first one is a reconstruction operator in the right-hand side that improves the correct balancing of gradient forces in hydrostatic situations. This so-called gradient-robustness makes the scheme asymptotics-preserving (AP) in the zero Mach number limit in the sense that it converges on fixed grids to a pressure-robust scheme for the incompressible Stokes problem. The second ingredient is an AP-compatible stabilization term in the continuity equation that allows a-priori provable strong convergence of the scheme, in particular the satisfaction of the nonlinear barotropic equation of state in the limit. The paper explains the algorithmical details, proves stability and convergence of the scheme as well as meaningful error estimates for the velocity. The results are illustrated by some numerical examples

Combined effects of evaporation, sedimentation and solute crystallization on the dynamics of aerosol size distributions on multiple length and time scales

We investigate three aspects of aerosol-mediated air-borne viral infection mechanisms on different length and time scales. First, we address the evolution of the size distribution of a non-interacting ensemble of droplets that are subject to evaporation and sedimentation using a sharp droplet-air interface model. From the exact solution of the evolution equation we derive the viral load in the air and show that it depends sensitively on the relative humidity. Secondly, from Molecular Dynamics simulations we extract the molecular reflection coefficient of single water molecules from the air-water interface. This parameter determines the water condensation and evaporation rate at a liquid droplet surface and therefore the evaporation rate of aqueous droplets. We find the reflection of water to be negligible at room temperature but to rise significantly at elevated temperatures and for grazing incidence angles. Thirdly, we derive a thermodynamically consistent three-dimensional diffuse-interface model for solute-containing droplets that is formulated as a three-phase Cahn--Hilliard/Allen--Cahn system. By numerically solving the coupled system of equations, we explore representative scenarios that show that this model reproduces and generalizes features of the sharp-interface model. These interconnected studies on the dynamics of aerosol droplet evaporation are relevant in order to quantitatively assess the airborne infection risk under varying environmental conditions

Hybrid Machine learning based Scale Bridging Framework for Permeability Prediction of Fibrous Structures

This study introduces a hybrid machine learning-based scale-bridging framework for predicting the permeability of fibrous textile structures. By addressing the computational challenges inherent to multiscale modeling, the proposed approach evaluates the efficiency and accuracy of different scale-bridging methodologies combining traditional surrogate models and even integrating physics-informed neural networks (PINNs) with numerical solvers, enabling accurate permeability predictions across micro- and mesoscales. Four methodologies were evaluated: fully resolved models (FRM), numerical upscaling method (NUM), scale-bridging method using data-driven machine learning methods (SBM) and a hybrid dual-scale solver incorporating PINNs. The FRM provides the highest fidelity model by fully resolving the micro- and mesoscale structural geometries, but requires high computational effort. NUM is a fully numerical dual-scale approach that considers uniform microscale permeability but neglects the microscale structural variability. The SBM accounts for the variability through a segment-wise assigned microscale permeability, which is determined using the data-driven ML method. This method shows no significant improvements over NUM with roughly the same computational efficiency and modeling runtimes of 45 min per simulation. The newly developed hybrid dual-scale solver incorporating PINNs shows the potential to overcome the problem of data scarcity of the data-driven surrogate approaches, as well as incorporating data from both scales via the hybrid loss function. The hybrid framework advances permeability modeling by balancing computational cost and prediction reliability, laying the foundation for further applications in fibrous composite manufacturing, while its full potential awaits realization as physics-informed machine learning approaches continue to mature

Optimal control for a fourth-order nonisothermal tumor growth model of Caginalp type

We study a distributed optimal control problem for a nonisothermal Caginalp-type phase-field model that describes tumour growth under thermal therapy. The PDE system couples a possibly viscous Cahn–Hilliard equation, governing the evolution of the healthy and tumor phases, with an equation for the heat balance, and a reaction-diffusion equation for the nutrient concentration. Chemotaxis and active transport effects are taken into account, and hyperthermia appears as a control variable. We introduce a suitable tracking-type cost functional and show the existence of optimal controls. Then, we analyse the differentiability of the control-to-state operator and establish necessary first-order conditions expressed through a variational inequality involving the adjoint state variables