Marius Zeinhofer

Simula Research Laboratory

We propose energy natural gradient descent, a natural gradient method with respect to a Hessian-induced Riemannian metric as an optimization algorithm for physics-informed neural networks (PINNs) and the deep Ritz method. As a main motivation we show that the update direction in function space resulting from the energy natural gradient corresponds...

We analyze the training process of the Deep Ritz Method for elliptic equations with Dirichlet boundary conditions and highlight problems arising from essential boundary values. Typically, one employs a penalty approach to enforce essential boundary conditions, however, the naive approach to this problem becomes unstable for large penalizations. A n...

We provide convergence guarantees for the Deep Ritz Method for abstract variational energies. Our results cover nonlinear variational problems such as the p -Laplace equation or the Modica–Mortola energy with essential or natural boundary conditions. Under additional assumptions, we show that the convergence is uniform across bounded families of ri...

We establish error estimates for the approximation of parametric $p$-Dirichlet problems deploying the Deep Ritz Method. Parametric dependencies include, e.g., varying geometries and exponents $p\in (1,\infty)$. Combining the derived error estimates with quantitative approximation theorems yields error decay rates and establishes that the Deep Ritz...

We present a three dimensional, homogenized PDE/ODE model for bone fracture healing in the presence of a porous, bio-resorbable scaffold and an associated PDE constrained optimization problem concerning the optimal scaffold density distribution for an ideal healing environment. The model is analyzed mathematically and a well-posedness result is pro...

We consider the scaffold design optimization problem associated to the three dimensional, time dependent model for scaffold mediated bone regeneration considered in Dondl et al. (2021). We prove existence of optimal scaffold designs and present numerical evidence that optimized scaffolds mitigate stress shielding effects from exterior fixation of t...

We prove an $L^p(I,C^\alpha(\Omega))$ regularity result for a reaction-diffusion equation with mixed boundary conditions, symmetric $L^\infty$ coefficients and an $L^\infty$ initial condition. We provide explicit control of the $L^p(I,C^\alpha(\Omega))$ norm with respect to the data. To prove our result, we first establish $C^\alpha(\Omega)$ contro...

We provide convergence guarantees for the Deep Ritz Method for abstract variational energies. Our results cover non-linear variational problems such as the $p$-Laplace equation or the Modica-Mortola energy with essential or natural boundary conditions. Under additional assumptions, we show that the convergence is uniform across % bounded families o...

We propose a simple model for scaffold aided bone regeneration. In this model, only macroscopic quantities, e.g., locally averaged osteoblast densities, are considered. This allows for use of this model in an optimization algorithm, whose outcome is an optimal scaffold porosity distribution. This optimal scaffold naturally depends on the choice of...

We compare different training strategies for the Deep Ritz Method for elliptic equations with Dirichlet boundary conditions and highlight the problems arising from the boundary values. We distinguish between an exact resolution of the boundary values by introducing a distance function and the approximation through a Robin Boundary Value problem. Ho...

We analyse the difference in convergence mode using exact versus penalised boundary values for the residual minimisation of PDEs with neural network type ansatz functions, as is commonly done in the context of physics informed neural networks. It is known that using an $L^2$ boundary penalty leads to a loss of regularity of $3/2$ meaning that appro...

We establish estimates on the error made by the Ritz method for quadratic energies on the space $H^1(\Omega)$ in the approximation of the solution of variational problems with different boundary conditions. Special attention is paid to the case of Dirichlet boundary values which are treated with the boundary penalty method. We consider arbitrary an...

We present a three dimensional, time dependent model for bone regeneration in the presence of porous scaffolds to bridge critical size bone defects. Our approach uses homogenized quantities, thus drastically reducing computational cost compared to models resolving the microstructural scale of the scaffold. Using abstract functional relationships in...

Recently, progress has been made in the application of neural networks to the numerical analysis of partial differential equations (PDEs). In the latter the variational formulation of the Poisson problem is used in order to obtain an objective function - a regularised Dirichlet energy - that was used for the optimisation of some neural networks. In...