Peter Virsik presents his B.Sc. thesis on "Surrogates for Learning Input to Solution Map for the 1D Parametric (Transient) Heat Equation by Using Weighted Residuals"

Abstract:
Heat transfer simulations are an essential tool in engineering. Traditionally, temperature distributions in a body are computed by solving the heat equation numerically with a fi-
nite element method (FEM). However, when many simulations for different problem variants (e.g. for testing different materials) are required, this approach is expensive and thus not scalable. Recent advances in deep neural network based surrogates have enabled models that can learn solutions of partial differential equations (PDEs) for varying input parameters.
In this work, surrogates are developed to approximate solutions of the heat equation for different materials. First, solutions of the 1D steady-state heat equation are approximated based on the material’s conductivity field. Second, a surrogate is developed to map the volumetric heat capacity and the conductivity field of the material to the solution of the 1D transient heat equation. The proposed physics informed surrogate uses weighted residuals rather than probing the heat equation at a set of collocation points. In order to represent the full solution field the surrogate uses Bernstein polynomials as shape functions. To improve training stability and speed the network is designed such that the boundary and initial conditions are satisfied by construction. The capabilities of the developed framework are demonstrated with an optimization problem. The optimization considers the design of an optimal heat insulation. By combining the surrogate with gradient-based optimization, optimal material fieldscan be determined efficiently.