arXiv subjects

Sebastian K. Mitusch

Publications and source records attributed to Sebastian K. Mitusch.

3 recordsLinked to original sources

arXiv2021

Hybrid FEM-NN models: Combining artificial neural networks with the finite element method

We present a methodology combining neural networks with physical principle constraints in the form of partial differential equations (PDEs). The approach allows to train neural networks while respecting the PDEs as a strong constraint in the optimisation as apposed to making them part of the loss function. The resulting models are discretised in space by the finite element method (FEM). The method applies to both stationary and transient as well as linear/nonlinear PDEs. We describe implementation of the approach as an extension of the existing FEM framework FEniCS and its algorithmic differentiation tool dolfin-adjoint. Through series of examples we demonstrate capabilities of the approach to recover coefficients and missing PDE operators from observations. Further, the proposed method is compared with alternative methodologies, namely, physics informed neural networks and standard PDE-constrained optimisation. Finally, we demonstrate the method on a complex cardiac cell model problem using deep neural networks.

math.NA↗

arXiv2020

Automatic shape derivatives for transient PDEs in FEniCS and Firedrake

In industry, shape optimization problems are of utter importance when designing structures such as aircraft, automobiles and turbines. For many of these applications, the structure changes over time, with a prescribed or non-prescribed movement. Therefore, it is important to capture these features in simulations when optimizing the design of the structure. Using gradient based algorithms, deriving the shape derivative manually can become very complex and error prone, especially in the case of time-dependent non-linear partial differential equations. To ease this burden, we present a high-level algorithmic differentiation tool that automatically computes first and second order shape derivatives for partial differential equations posed in the finite element frameworks FEniCS and Firedrake. The first order shape derivatives are computed using the adjoint method, while the second order shape derivatives are computed using a combination of the tangent linear method and the adjoint method. The adjoint and tangent linear equations are symbolically derived for any sequence of variational forms. As a consequence our methodology works for a wide range of PDE problems and is discretely consistent. We illustrate the generality of our framework by presenting several examples, spanning the range of linear, non-linear and time-dependent PDEs for both stationary and transient domains.

math.OC↗

arXiv2019

Can diffusion alone explain brain-wide distribution of a CSF tracer within 24 hours?

The recently proposed glymphatic system suggests that bulk flow is important for clearing waste from the brain, and as such may underlie the development of e.g. Alzheimer's disease. The glymphatic hypothesis is still controversial and several biomechanical modeling studies at the micro-level have at least partially dismissed the system and its assumptions. In contrast, at the macro-level, there are many experimental findings in support of bulk flow. Here, we will investigate to what extent the CSF tracer distributions seen in novel magnetic resonance imaging (MRI) investigations over hours and days are suggestive of bulk flow or diffusion. In order to include the complex geometry of the brain, the heterogeneous CSF flow around the brain, and the transport over the time-scale of days, we employed the methods of partial differential constrained optimization to identify the apparent diffusion coefficient (ADC) that would correspond best to the MRI findings. We found that the computed ADC in grey and white matter was respectively 23\% and 82\% larger than the ADC estimated with DTI, which suggests that diffusion may not be the only mechanism governing transport.

math.NA↗