arXiv subjects

Gregor Maier

Publications and source records attributed to Gregor Maier.

4 recordsLinked to original sources

arXiv2026

Near-Optimal Learning of Gaussian Sobolev Operators

A key question in operator learning is how to design surrogate operators with provable approximation guarantees in reasonable computational time. Whereas smooth operators can be approximated efficiently, i.e., with at least algebraic convergence in the amount of training data, learning finitely regular operators is known to be less efficient. The reason is an intrinsic curse of sample complexity, which allows only subalgebraic sample complexity rates. This fact makes it all the more important to develop algorithms which provably achieve these rates. In this work, we present a fully data-driven algorithm, termed Hermite-PCA approximation, for learning Gaussian Sobolev operators with near-optimal sample complexity. It employs principal component analysis and weighted least-squares methods and is therefore computationally efficient. Moreover, it is spectral, in the sense that it achieves faster (and near-optimal) convergence the higher the Sobolev regularity. We provide a full error analysis of this algorithm, taking into account all sources of error, along with numerical experiments that verify our theoretical results and empirically confirm the efficacy of Hermite-PCA approximation for learning Sobolev operators.

math.NA↗

arXiv2025

Towards Sharp Minimax Risk Bounds for Operator Learning

We develop a minimax theory for operator learning, where the goal is to estimate an unknown operator between separable Hilbert spaces from finitely many noisy input-output samples. For uniformly bounded Lipschitz operators, we prove information-theoretic lower bounds together with matching or near-matching upper bounds, covering both fixed and random designs under Hilbert-valued Gaussian noise and Gaussian white noise errors. The rates are controlled by the spectrum of the covariance operator of the measure that defines the error metric. Our setup is very general and allows for measures with unbounded support. A key implication is a curse of sample complexity, which shows that the minimax risk for generic Lipschitz operators cannot decay at any algebraic rate in the sample size. We obtain sharp characterizations when the covariance spectrum decays exponentially and provide general upper and lower bounds in slower-decay regimes. Finally, we show that assuming higher regularity, i.e., H\"older smoothness, does not improve minimax rates over the Lipschitz case, up to potential constants. Thus, we show that learning operators of any finite regularity necessarily suffers a curse of sample complexity.

math.ST↗

arXiv2024

The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measures

Operator learning, the approximation of mappings between infinite-dimensional function spaces using machine learning, has gained increasing research attention in recent years. Operator approximations can serve as efficient surrogate models for problems in computational science and engineering, complementing traditional methods. However, despite their empirical success, our understanding of the underlying mathematical theory is in large part still incomplete. In this paper, we study the approximation of Lipschitz operators with respect to Gaussian measures. We prove higher Gaussian Sobolev regularity of Lipschitz operators and establish lower and upper bounds on the Hermite polynomial approximation error. We then study general reconstruction strategies of Lipschitz operators from $m$ arbitrary (potentially adaptive) linear samples. As a key finding, we tightly characterize the corresponding sample complexity, that is, the smallest achievable worst-case error among all possible choices of (adaptive) sampling and reconstruction strategies, in terms of $m$. As a consequence, we identify an inherent curse of sample complexity: No method to approximate Lipschitz operators based on $m$ linear samples can achieve algebraic convergence rates in $m$. On the positive side, we prove that a sufficiently fast spectral decay of the covariance operator of the underlying Gaussian measure guarantees convergence rates which are arbitrarily close to any algebraic rate. Overall, by tightly characterizing the sample complexity, our work confirms the intrinsic difficulty of learning Lipschitz operators, regardless of the data or learning technique.

cs.LG↗

arXiv2023

Predicting Properties of Oxide Glasses Using Informed Neural Networks

Many modern-day applications require the development of new materials with specific properties. In particular, the design of new glass compositions is of great industrial interest. Current machine learning methods for learning the composition-property relationship of glasses promise to save on expensive trial-and-error approaches. Even though quite large datasets on the composition of glasses and their properties already exist (i.e., with more than 350,000 samples), they cover only a very small fraction of the space of all possible glass compositions. This limits the applicability of purely data-driven models for property prediction purposes and necessitates the development of models with high extrapolation power. In this paper, we propose a neural network model which incorporates prior scientific and expert knowledge in its learning pipeline. This informed learning approach leads to an improved extrapolation power compared to blind (uninformed) neural network models. To demonstrate this, we train our models to predict three different material properties, that is, the glass transition temperature, the Young's modulus (at room temperature), and the shear modulus of binary oxide glasses which do not contain sodium. As representatives for conventional blind neural network approaches we use five different feed-forward neural networks of varying widths and depths. For each property, we set up model ensembles of multiple trained models and show that, on average, our proposed informed model performs better in extrapolating the three properties of previously unseen sodium borate glass samples than all five conventional blind models.

physics.comp-ph↗