Paul Schwerdtner

Paul Schwerdtner, Tobias Blickhan, Benjamin Peherstorfer

This work addresses the problem of learning the dynamics of high-dimensional probability densities over time using unlabeled samples, without assuming access to trajectory information. We introduce two-parameter flows that learn only sampling-time transports from a base distribution to each marginal and then extract a physics-time velocity by regressing on coupled synthetic trajectories. We prove that the resulting physics-time dynamics are unique and inherit regularity from the sampling-time transports. Because we can build on standard, well-developed conditional flow matching techniques for learning the base-to-marginal transports, our approach scales to high dimensions and avoids per-step optimal-transport couplings, while allowing admissible non-gradient dynamics that can naturally explain rotational or circulating physics phenomena.

Paul Schwerdtner, Philipp Schulze, Jules Berman et al.

This work focuses on the conservation of quantities such as Hamiltonians, mass, and momentum when solution fields of partial differential equations are approximated with nonlinear parametrizations such as deep networks. The proposed approach builds on Neural Galerkin schemes that are based on the Dirac--Frenkel variational principle to train nonlinear parametrizations sequentially in time. We first show that only adding constraints that aim to conserve quantities in continuous time can be insufficient because the nonlinear dependence on the parameters implies that even quantities that are linear in the solution fields become nonlinear in the parameters and thus are challenging to discretize in time. Instead, we propose Neural Galerkin schemes that compute at each time step an explicit embedding onto the manifold of nonlinearly parametrized solution fields to guarantee conservation of quantities. The embeddings can be combined with standard explicit and implicit time integration schemes. Numerical experiments demonstrate that the proposed approach conserves quantities up to machine precision.

Paul Schwerdtner, Florens Greßner, Nikhil Kapoor et al.

In this paper we propose a framework for assessing the risk associated with deploying a machine learning model in a specified environment. For that we carry over the risk definition from decision theory to machine learning. We develop and implement a method that allows to define deployment scenarios, test the machine learning model under the conditions specified in each scenario, and estimate the damage associated with the output of the machine learning model under test. Using the likelihood of each scenario together with the estimated damage we define \emph{key risk indicators} of a machine learning model. The definition of scenarios and weighting by their likelihood allows for standardized risk assessment in machine learning throughout multiple domains of application. In particular, in our framework, the robustness of a machine learning model to random input corruptions, distributional shifts caused by a changing environment, and adversarial perturbations can be assessed.

Nicat Aliyev, Peter Benner, Emre Mengi et al.

We are concerned with the computation of the ${\mathcal L}_\infty$-norm for an ${\mathcal L}_\infty$-function of the form $H(s) = C(s) D(s)^{-1} B(s)$, where the middle factor is the inverse of a meromorphic matrix-valued function, and $C(s),\, B(s)$ are meromorphic functions mapping to short-and-fat and tall-and-skinny matrices, respectively. For instance, transfer functions of descriptor systems and delay systems fall into this family. We focus on the case where the middle factor is large-scale. We propose a subspace projection method to obtain approximations of the function $H$ where the middle factor is of much smaller dimension. The ${\mathcal L}_\infty$-norms are computed for the resulting reduced functions, then the subspaces are refined by means of the optimal points on the imaginary axis where the ${\mathcal L}_\infty$-norm of the reduced function is attained. The subspace method is designed so that certain Hermite interpolation properties hold between the largest singular values of the original and reduced functions. This leads to a locally superlinearly convergent algorithm with respect to the subspace dimension, which we prove and illustrate on various numerical examples.