We study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating multi scale PDEs. The latter may arise in multi scale material models where, e.g. in a two-scale setting, the coarse scale model is accessible via the FEM whereas the fine scale problem is approached by a NN technique. In fact, from a continuous perspective, our formulation corresponds to a non-standard PDE-constrained optimization problem with a PINN-type objective. From a discrete standpoint, the formulation represents a hybrid numerical solver that utilizes both neural networks and finite elements. We propose a function space framework for the problem and develop an algorithm for its numerical solution, combining an adjoint-based technique from optimal control with automatic differentiation. As a proof of concept, our multi scale solver is applied to a heat transfer problem with oscillating coefficients, where the neural network approximates a fine-scale problem, and a coarse-scale problem constrains the learning process. We show that incorporating coarse-scale information into the neural network training process through our modelling framework acts as a preconditioner for the low-frequency component of the fine-scale PDE, resulting in improved convergence properties and accuracy of the PINN method. The relevance of the hybrid solver to numerical homogenization is discussed.

We are interested in solving mathematical programs that feature distributed and discrete-valued decision variables that allow to control a dynamical system that is modeled by means of a PDE constraint. The distributed decision variables are measurable functions and regularized with a total variation penalty, that is the value of the regularizer is the sum of the interface areas of the level sets of decision variables weighted by the jump heights over the respective interfaces. In particular, the level sets have finite perimeters. In the multi-dimensional case, current discrete optimization-based techniques enable us to solve medium-sized problem instances to optimality in an acceptable amount of time in the context of descent algorithms. Realistic problem sizes may currently be deemed intractable, however. In order to overcome this issue, we propose and analyze a domain decomposition technique that allows to replace the solution of large-scale problem instances to many medium-sized problems that are coupled through overlaps of the subdomains on which the smaller problems are defined. To this end, we employ covering arguments inside of an algorithmic framework to localize the optimality and sufficient decrease conditions for a superordinate trust-region strategy. We employ an algorithm leans on greedy approaches in coordinate descent algorithms in order to obtain convergence to stationary points.

We consider subproblems arising from a trust-region algorithm for mixed integer optimal control problems with total variation regularization and integer-valued control functions. After discretization we obtain a class of integer linear programs. We discuss the NP-hardness for the one-dimensional and the two-dimensional case. In the one-dimensional case, the subproblems can be transformed into shortest path problems from a source to a sink on a layered graph. This graph-based approach allows us to solve the one-dimensional problem in pseudo-polynomial time. We shortly discuss why this approach does not work for the two-dimensional case.
Instead, we observe that the underlying polyhedrons exhibit structural restrictions in the vertices regarding which variables can attain fractional values at the same time. We use a graph-based view of the problem and this special property of the underlying polyhedron to obtain valid cutting planes to tighten the linear programming relaxation. Furthermore, we briefly discuss how good primal solutions and a branching rule can be derived from the one-dimensional case.

We consider integer optimal control problems with total variation regularization on multi-dimensional domains and derive first-order optimality conditions by means of local variations of the level sets of the feasible control functions. In order to solve this class of problems, we present a function space trust-region algorithm as well as an outer approximation algorithm to solve such problems after finite-element discretization. The control functions are discretized with integer-valued functions that are constant on the cells of a cubic mesh. We introduce a discretized total variation whose discretization is coupled to but varying from the discretization of the control functions. The coupling of the discretizations allows to recover the total variation of discontinuous integer-valued control functions with the integer-valued discretized control functions in the sense of Gamma-convergence. We add a constraint to the discretized problems that vanishes in the limit and that enforces compactness which guarantees the existence of minimizers. We prove that those minimizers converge to minimizers of the original problem when the mesh sizes of the coupled discretizations are driven to zero. We will conclude by presenting some numerical examples that confirm our theoretical results.