H. Harder

Sebastian Peitz, Hans Harder, Feliks Nüske et al.

The Koopman operator has become an essential tool for data-driven analysis, prediction and control of complex systems. The main reason is the enormous potential of identifying linear function space representations of nonlinear dynamics from measurements. This equally applies to ordinary, stochastic, and partial differential equations (PDEs). Until now, with a few exceptions only, the PDE case is mostly treated rather superficially, and the specific structure of the underlying dynamics is largely ignored. In this paper, we show that symmetries in the system dynamics can be carried over to the Koopman operator, which allows us to significantly increase the model efficacy. Moreover, the situation where we only have access to partial observations (i.e., measurements, as is very common for experimental data) has not been treated to its full extent, either. Moreover, we address the highly-relevant case where we cannot measure the full state, where alternative approaches (e.g., delay coordinates) have to be considered. We derive rigorous statements on the required number of observables in this situation, based on embedding theory. We present numerical evidence using various numerical examples including the wave equation and the Kuramoto-Sivashinsky equation.

Hans Harder, Abhijeet Vishwasrao, Luca Guastoni et al.

This paper is concerned with probabilistic techniques for forecasting dynamical systems described by partial differential equations (such as, for example, the Navier-Stokes equations). In particular, it is investigating and comparing various extensions to the flow matching paradigm that reduce the number of sampling steps. In this regard, it compares direct distillation, progressive distillation, adversarial diffusion distillation, Wasserstein GANs and rectified flows. Moreover, experiments are conducted on a set of challenging systems. In particular, we also address the challenge of directly predicting 2D slices of large-scale 3D simulations, paving the way for efficient inflow generation for solvers.

Jan Stenner, Hans Harder, Sebastian Peitz

This paper studies sparse sensor placement for control of Rayleigh-Bénard convection with multi-agent reinforcement learning. We train dense expert policies with windowed observations and distill sparse apprentice policies by supervised learning with grouped regularization on encoder input weights. The framework combines ordered non-convex grouped regularization and iterative reweighted grouped regularization, and uses a grouping construction that enforces consistent pruning across overlapping observation windows. Experiments with fixed and varying initial conditions show that Multi-Agent Transformer policies train more stably than proximal policy optimization baselines, while sparse apprentices retain control behavior comparable to dense experts. Sparsity results are strong for the proposed grouped methods across settings, including maximal sparsity in all fixed-initial-condition setting variants and maximal or near-maximal sparsity in varying-initial-condition setting variants. As an additional proof of concept, training from learned minimal sensor sets reduces per-agent observation size from 360 to 12 and preserves the overall training trend in simulation while reducing data throughput. The results provide both an interpretable basis for identifying control-relevant spatial regions and state components, and a practical pathway toward sensor-efficient control under realistic hardware constraints.

Hans Harder, Jean Rabault, Ricardo Vinuesa et al.

We utilize extreme-learning machines for the prediction of partial differential equations (PDEs). Our method splits the state space into multiple windows that are predicted individually using a single model. Despite requiring only few data points (in some cases, our method can learn from a single full-state snapshot), it still achieves high accuracy and can predict the flow of PDEs over long time horizons. Moreover, we show how additional symmetries can be exploited to increase sample efficiency and to enforce equivariance.

Hans Harder, Sebastian Peitz

The value function plays a crucial role as a measure for the cumulative future reward an agent receives in both reinforcement learning and optimal control. It is therefore of interest to study how similar the values of neighboring states are, i.e., to investigate the continuity of the value function. We do so by providing and verifying upper bounds on the value function's modulus of continuity. Additionally, we show that the value function is always Hölder continuous under relatively weak assumptions on the underlying system and that non-differentiable value functions can be made differentiable by slightly "disturbing" the system.