arXiv subjects

Adrien Weihs

Publications and source records attributed to Adrien Weihs.

10 recordsLinked to original sources

arXiv2026

Kernel Methods for Learning Operators with Multiple Inputs and Outputs

Learning mappings between infinite-dimensional objects is a central challenge in scientific machine learning. We introduce a general kernel-based encoder-decoder framework for operator learning that separates observation, representation, learning, and reconstruction. We develop this framework for multi-input, multi-output operator learning, where operators map between products of potentially distinct function spaces. Our approximation theory shows that, although the number of inputs and outputs can increase, the convergence rate is governed by the most challenging constituent approximation problem rather than the overall problem dimension. The framework leads to practical kernel methods with closed-form training and inference, combining mathematical tractability with computational efficiency. We further specialize the approach to multiple operator learning by introducing KernelMO, a family of kernel methods with complementary operator-valued and product-space formulations. Across five families of parametric partial differential equations, the proposed methods achieve competitive or state-of-the-art predictive accuracy while reducing training and inference costs relative to neural operator architectures and deep learning based models, offering an efficient and lightweight alternative.

cs.LG↗

arXiv2026

Multiple Neural Operators Achieve Near-Optimal Rates for Multi-Task Learning

We study the approximation and statistical complexity of learning collections of operators in a shared multi-task setting, with a focus on the Multiple Neural Operators (MNO) architecture. For broad classes of Lipschitz multiple operator maps, we derive near-optimal upper bounds for approximation and statistical generalization. On the lower-bound side, we establish a curse of parametric complexity and prove corresponding minimax rates. Together, these results show that shared representations across tasks do not increase the overall cost: multi-task operator learning follows the same scaling laws as single operator learning. We also compare MNO with a multi-task extension of DeepONet based on concatenated task inputs and show that, from a worst-case approximation-complexity perspective, both architectures satisfy essentially the same asymptotic rates.

cs.LG↗

arXiv2026

Generalization Bounds and Statistical Guarantees for Multi-Task and Multiple Operator Learning with MNO Networks

Multiple operator learning concerns learning operator families $\{G[\alpha]:U\to V\}_{\alpha\in W}$ indexed by an operator descriptor $\alpha$. Training data are collected hierarchically by sampling operator instances $\alpha$, then input functions $u$ per instance, and finally evaluation points $x$ per input, yielding noisy observations of $G[\alpha][u](x)$. While recent work has developed expressive multi-task and multiple operator learning architectures and approximation-theoretic scaling laws, quantitative statistical generalization guarantees remain limited. We provide a covering-number-based generalization analysis for separable models, focusing on the Multiple Neural Operator (MNO) architecture: we first derive explicit metric-entropy bounds for hypothesis classes given by linear combinations of products of deep ReLU subnetworks, and then combine these complexity bounds with approximation guarantees for MNO to obtain an explicit approximation-estimation tradeoff for the expected test error on new (unseen) triples $(\alpha,u,x)$. The resulting bound makes the dependence on the hierarchical sampling budgets $(n_\alpha,n_u,n_x)$ transparent and yields an explicit learning-rate statement in the operator-sampling budget $n_\alpha$, providing a sample-complexity characterization for generalization across operator instances. The structure and architecture can also be viewed as a general purpose solver or an example of a "small'' PDE foundation model, where the triples are one form of multi-modality.

cs.LG↗

arXiv2025

Higher-Order Regularization Learning on Hypergraphs

Higher-Order Hypergraph Learning (HOHL) was recently introduced as a principled alternative to classical hypergraph regularization, enforcing higher-order smoothness via powers of multiscale Laplacians induced by the hypergraph structure. Prior work established the well- and ill-posedness of HOHL through an asymptotic consistency analysis in geometric settings. We extend this theoretical foundation by proving the consistency of a truncated version of HOHL and deriving explicit convergence rates when HOHL is used as a regularizer in fully supervised learning. We further demonstrate its strong empirical performance in active learning and in datasets lacking an underlying geometric structure, highlighting HOHL's versatility and robustness across diverse learning settings.

cs.LG↗

arXiv2025

Analysis of Semi-Supervised Learning on Hypergraphs

Hypergraphs provide a natural framework for modeling multiway interactions. We analyze a class of variational semi-supervised learning problems posed on random geometric hypergraphs and establish asymptotic consistency in the large-data limit. In particular, we identify scaling regimes that ensure well-posedness--yielding nontrivial label propagation rather than collapse to a constant labeling--and show that discrete minimizers converge, in the continuum, to solutions of a density-weighted p-Laplacian equation. We also propose Higher-Order Hypergraph Learning (HOHL), a multiscale regularization scheme based on powers of Laplacians associated with hypergraph-induced subgraphs. For geometric point clouds, we analyze an efficient multiscale Laplacian surrogate for HOHL and prove convergence to a higher-order Sobolev-type seminorm. Numerical experiments on standard benchmarks support the practical utility of the resulting higher-order regularization.

cs.LG↗

arXiv2025

A Deep Learning Framework for Multi-Operator Learning: Architectures and Approximation Theory

While many problems in machine learning focus on learning mappings between finite-dimensional spaces, scientific applications require approximating mappings between function spaces, i.e., operators. We study the problem of learning collections of operators and provide both theoretical and empirical advances. We distinguish between two regimes: (i) multiple operator learning, where a single network represents a continuum of operators parameterized by a parametric function, and (ii) learning several distinct single operators, where each operator is learned independently. For the multiple operator case, we introduce two new architectures, $\mathrm{MNO}$ and $\mathrm{MONet}$, and establish universal approximation results in three settings: continuous, integrable, or Lipschitz operators. For the latter, we further derive explicit scaling laws that quantify how the network size must grow to achieve a target approximation accuracy. For learning several single operators, we develop a framework for balancing architectural complexity across subnetworks and show how approximation order determines computational efficiency. Empirical experiments on parametric PDE benchmarks confirm the strong expressive power and efficiency of the proposed architectures. Overall, this work establishes a unified theoretical and practical foundation for scalable neural operator learning across multiple operators.

cs.LG↗

arXiv2025

Topology-Aware Active Learning on Graphs

We propose a graph-topological approach to active learning that directly targets the core challenge of exploration versus exploitation under scarce label budgets. To guide exploration, we introduce a coreset construction algorithm based on Balanced Forman Curvature (BFC), which selects representative initial labels that reflect the graph's cluster structure. This method includes a data-driven stopping criterion that signals when the graph has been sufficiently explored. We further use BFC to dynamically trigger the shift from exploration to exploitation within active learning routines, replacing hand-tuned heuristics. To improve exploitation, we introduce a localized graph rewiring strategy that efficiently incorporates multiscale information around labeled nodes, enhancing label propagation while preserving sparsity. Experiments on benchmark classification tasks show that our methods consistently outperform existing graph-based semi-supervised baselines at low label rates.

cs.LG↗

arXiv2025

Unsupervised and semi-supervised clustering methods to identify and refine participant experience levels in educational research

The progression from novice to disciplinary expert is a longstanding area of inquiry in educational research. Studies investigating such progressions have often resorted to participants' self-assessments or other qualitative indicators as a starting point to define experience. But does a participant's estimated experience coincide with metrics derived from their conceptual understanding of a discipline? Using data extracted from over 150 concept maps, we first demonstrate that disciplinary experience is a reliable variable to explain differences in conceptual understanding across a highly diverse learners' population. Through a comparison of unsupervised and semi-supervised models, we then motivate clustering participants into three distinguished experience levels, and support such a classification performed in other studies of educational research. By analysing cluster composition, we also identify discrepancies between the perceived and predicted experience levels of the study participants. Lastly, for studies processing participants data through network analysis, we present insights into statistically significant metrics that can characterise each experience level, and advocate for the use of node-level metrics in such studies.

physics.ed-ph↗

arXiv2023

Discrete-to-Continuum Rates of Convergence for $p$-Laplacian Regularization

Higher-order regularization problem formulations are popular frameworks used in machine learning, inverse problems and image/signal processing. In this paper, we consider the computational problem of finding the minimizer of the Sobolev $\mathrm{W}^{1,p}$ semi-norm with a data-fidelity term. We propose a discretization procedure and prove convergence rates between our numerical solution and the target function. Our approach consists of discretizing an appropriate gradient flow problem in space and time. The space discretization is a nonlocal approximation of the p-Laplacian operator and our rates directly depend on the localization parameter $ε_n$ and the time mesh-size $τ_n$. We precisely characterize the asymptotic behaviour of $ε_n$ and $τ_n$ in order to ensure convergence to the considered minimizer. Finally, we apply our results to the setting of random graph models.

math.NA↗

arXiv2023

Consistency of Fractional Graph-Laplacian Regularization in Semi-Supervised Learning with Finite Labels

Laplace learning is a popular machine learning algorithm for finding missing labels from a small number of labelled feature vectors using the geometry of a graph. More precisely, Laplace learning is based on minimising a graph-Dirichlet energy, equivalently a discrete Sobolev $\Wkp{2}{1}$ semi-norm, constrained to taking the values of known labels on a given subset. The variational problem is asymptotically ill-posed as the number of unlabeled feature vectors goes to infinity for finite given labels due to a lack of regularity in minimisers of the continuum Dirichlet energy in any dimension higher than one. In particular, continuum minimisers are not continuous. One solution is to consider higher-order regularisation, which is the analogue of minimising Sobolev $\Wkp{s}{2}$ semi-norms. In this paper we consider the asymptotics of minimising a graph variant of the Sobolev $\Wkp{s}{2}$ semi-norm with pointwise constraints. We show that, as expected, one needs $s>d/2$ where $d$ is the dimension of the data manifold. We also show that there must be an upper bound on the connectivity of the graph; that is, highly connected graphs lead to degenerate behaviour of the minimiser even when $s>d/2$.

math.ST↗