Lars Holmberg

Lars Holmberg is a Senior Associate in the Numerical Methods and Computational Mathematics Division at the Institute for Advanced Dynamic Uncertainty, where his research centres on the use of deep learning to solve high-dimensional partial differential equations arising in stochastic control, mean field games, and mathematical finance. He holds a PhD in Applied Mathematics from Chalmers University of Technology (Department of Mathematical Sciences), where his doctoral work introduced neural network approximations for the value function of high-dimensional Hamilton–Jacobi–Bellman equations, establishing convergence rates for the deep Galerkin method applied to parabolic control problems with up to 200 state dimensions. His thesis combined functional analytic tools — Sobolev space theory, energy estimates, variational formulations — with expressivity results from deep learning theory to derive error bounds that depend polynomially rather than exponentially on dimension, providing a rigorous approximation-theoretic foundation for neural PDE solvers in the optimal control setting.

Holmberg's subsequent work extended these methods to physics-informed neural networks (PINNs), where the PDE residual is incorporated directly into the loss function and training proceeds without labelled solution data. His research in this period addressed the pathological loss landscapes of PINN training on stiff or degenerate problems — situations common in financial PDEs — and he developed adaptive collocation strategies and gradient-weighting schemes that reliably recover sharp free boundaries in American option pricing and optimal stopping problems. He has contributed theoretical results on the trainability of deep networks for degenerate parabolic operators, with applications to Black–Scholes and local volatility PDEs in multiple asset dimensions.

At IADU, Holmberg leads the Institute's scientific machine learning programme, developing neural solvers for the coupled HJB–Fokker–Planck systems that arise in mean field game models of systemic risk, commodity markets, and sovereign debt dynamics. His implementations use PyTorch for automatic differentiation and custom training loops, DeepXDE for PINN-based PDE solving, and TensorFlow for production-grade neural operator architectures. He is responsible for integrating deep learning methods into IADU's broader PDE infrastructure, ensuring that high-dimensional control problems intractable under classical finite difference or finite element discretisation can be solved using neural approximation.