arXiv subjects

Lars Ruthotto

Publications and source records attributed to Lars Ruthotto.

At least 19 recordsLinked to original sources

arXiv2026

RAPNet: Accelerating Algebraic Multigrid with Learned Sparse Corrections

The scalable solution of large sparse linear systems is a bottleneck in scientific computing and graph analysis. While algebraic multigrid (AMG) offers optimal linear scaling, its performance is severely constrained by the trade-off between the sparsity and convergence quality of coarse-grid operators. Classical AMG heuristics struggle to balance these objectives, often sacrificing stability or performance for sparsity. We propose RAPNet, a graph neural network (GNN) framework that resolves this trade-off by learning to generate sparse, robust coarse operators directly from the sparse algebraic system. Key to our approach is a level-wise training strategy that enables learning from small subgraphs and generalization to million-node domains, bypassing the bottlenecks of prior neural AMG attempts. RAPNet executes exclusively during the solver setup phase, ensuring that the solve phase retains its favorable computational properties. We show that our method outperforms classical non-Galerkin baselines on diverse PDE discretizations and graph Laplacians, making it particularly effective for multi-query tasks such as eigenproblems, time-dependent simulations, and inverse or design problems.

cs.LG↗

arXiv2026

Manifold-Aware Perturbations for Constrained Generative Modeling

Generative models have enjoyed widespread success in a variety of applications. However, they encounter inherent mathematical limitations in modeling distributions where samples are constrained by equalities, as is frequently the setting in scientific domains. In this work, we develop a computationally cheap, mathematically justified, and highly flexible distributional modification for combating known pitfalls in equality-constrained generative models. We propose perturbing the data distribution in a constraint-aware way such that the new distribution has support matching the ambient space dimension while still implicitly incorporating underlying manifold geometry. Through theoretical analyses and empirical evidence on several representative tasks, we illustrate that our approach consistently enables data distribution recovery and stable sampling with both diffusion models and normalizing flows.

cs.LG↗

arXiv2026

Differentiating through Stochastic Differential Equations: A Primer

Dynamical systems are essential to model various phenomena in physics, finance, economics, and are also of current interest in machine learning. A central modeling task is investigating parameter sensitivity, whether tuning atmospheric coefficients, computing financial Greeks, or optimizing neural networks. These sensitivities are mathematically expressed as derivatives of an objective function with respect to parameters of interest and are rarely available analytically, necessitating numerical methods for approximating them. While the literature for differentiation of deterministic systems is well-covered, the treatment of stochastic systems, such as stochastic differential equations (SDEs), in most curricula is less comprehensive than the subtleties arising from the interplay of noise and discretization require. This paper provides a primer on numerical differentiation of SDEs organized as a two-tale narrative. Tale 1 demonstrates differentiating through discretized SDEs, known the discretize-optimize approach, is reliable for both It\^{o} and Stratonovich calculus. Tale 2 examines the optimize-discretize approach, investigating the continuous limit of backward equations from Tale 1 corresponding to the desired gradients. Our aim is to equip readers with a clear guide on the numerical differentiation of SDEs: computing gradients correctly in both It\^{o} and Stratonovich settings, understanding when discretize-optimize and optimize-discretize agree or diverge, and developing intuition for reasoning about stochastic differentiation beyond the cases explicitly covered.

math.NA↗

arXiv2025

Mixed Precision Training of Neural ODEs

Exploiting low-precision computations has become a standard strategy in deep learning to address the growing computational costs imposed by ever larger models and datasets. However, naively performing all computations in low precision can lead to roundoff errors and instabilities. Therefore, mixed precision training schemes usually store the weights in high precision and use low-precision computations only for whitelisted operations. Despite their success, these principles are currently not reliable for training continuous-time architectures such as neural ordinary differential equations (Neural ODEs). This paper presents a mixed precision training framework for neural ODEs consisting of explicit ODE solvers and a custom backpropagation scheme and shows their effectiveness in a range of learning tasks. Our scheme uses low-precision computations for evaluating the velocity, parameterized by the neural network, and for storing intermediate states, while numerical reliability is provided by custom dynamic adjoint scaling and by accumulating the solution and gradients in higher precision. These contributions address two key challenges in training neural ODEs: the computational cost of repeated network evaluations and the growth of memory requirements with the number of time steps or layers. Along with the paper we publish our extendable, open-source PyTorch package \texttt{rampde}, whose syntax resembles that of leading packages to provide a drop-in replacement in existing codes. We demonstrate the reliability and effectiveness of our scheme using challenging test cases and on neural ODE applications in image classification and generative models, achieving approximately 50\% memory reduction and up to 2x speedup while maintaining accuracy comparable to single-precision training.

cs.LG↗

arXiv2025

The Future of Artificial Intelligence and the Mathematical and Physical Sciences (AI+MPS)

This community paper developed out of the NSF Workshop on the Future of Artificial Intelligence (AI) and the Mathematical and Physics Sciences (MPS), which was held in March 2025 with the goal of understanding how the MPS domains (Astronomy, Chemistry, Materials Research, Mathematical Sciences, and Physics) can best capitalize on, and contribute to, the future of AI. We present here a summary and snapshot of the MPS community's perspective, as of Spring/Summer 2025, in a rapidly developing field. The link between AI and MPS is becoming increasingly inextricable; now is a crucial moment to strengthen the link between AI and Science by pursuing a strategy that proactively and thoughtfully leverages the potential of AI for scientific discovery and optimizes opportunities to impact the development of AI by applying concepts from fundamental science. To achieve this, we propose activities and strategic priorities that: (1) enable AI+MPS research in both directions; (2) build up an interdisciplinary community of AI+MPS researchers; and (3) foster education and workforce development in AI for MPS researchers and students. We conclude with a summary of suggested priorities for funding agencies, educational institutions, and individual researchers to help position the MPS community to be a leader in, and take full advantage of, the transformative potential of AI+MPS.

cs.AI↗

arXiv2025

Weight-Parameterization in Continuous Time Deep Neural Networks for Surrogate Modeling

Continuous-time deep learning models, such as neural ordinary differential equations (ODEs), offer a promising framework for surrogate modeling of complex physical systems. A central challenge in training these models lies in learning expressive yet stable time-varying weights, particularly under computational constraints. This work investigates weight parameterization strategies that constrain the temporal evolution of weights to a low-dimensional subspace spanned by polynomial basis functions. We evaluate both monomial and Legendre polynomial bases within neural ODE and residual network (ResNet) architectures under discretize-then-optimize and optimize-then-discretize training paradigms. Experimental results across three high-dimensional benchmark problems show that Legendre parameterizations yield more stable training dynamics, reduce computational cost, and achieve accuracy comparable to or better than both monomial parameterizations and unconstrained weight models. These findings elucidate the role of basis choice in time-dependent weight parameterization and demonstrate that using orthogonal polynomial bases offers a favorable tradeoff between model expressivity and training efficiency.

cs.LG↗

arXiv2024

PyHySCO: GPU-Enabled Susceptibility Artifact Distortion Correction in Seconds

Over the past decade, reversed Gradient Polarity (RGP) methods have become a popular approach for correcting susceptibility artifacts in Echo-Planar Imaging (EPI). Although several post-processing tools for RGP are available, their implementations do not fully leverage recent hardware, algorithmic, and computational advances, leading to correction times of several minutes per image volume. To enable 3D RGP correction in seconds, we introduce PyHySCO, a user-friendly EPI distortion correction tool implemented in PyTorch that enables multi-threading and efficient use of graphics processing units (GPUs). PyHySCO uses a time-tested physical distortion model and mathematical formulation and is, therefore, reliable without training. An algorithmic improvement in PyHySCO is its novel initialization scheme that uses 1D optimal transport. PyHySCO is published under the GNU public license and can be used from the command line or its Python interface. Our extensive numerical validation using 3T and 7T data from the Human Connectome Project suggests that PyHySCO achieves accuracy comparable to that of leading RGP tools at a fraction of the cost. We also validate the new initialization scheme, compare different optimization algorithms, and test the algorithm on different hardware and arithmetic precision.

math.OC↗

arXiv2024

Neural Network Approaches for Parameterized Optimal Control

We consider numerical approaches for deterministic, finite-dimensional optimal control problems whose dynamics depend on unknown or uncertain parameters. We seek to amortize the solution over a set of relevant parameters in an offline stage to enable rapid decision-making and be able to react to changes in the parameter in the online stage. To tackle the curse of dimensionality arising when the state and/or parameter are high-dimensional, we represent the policy using neural networks. We compare two training paradigms: First, our model-based approach leverages the dynamics and definition of the objective function to learn the value function of the parameterized optimal control problem and obtain the policy using a feedback form. Second, we use actor-critic reinforcement learning to approximate the policy in a data-driven way. Using an example involving a two-dimensional convection-diffusion equation, which features high-dimensional state and parameter spaces, we investigate the accuracy and efficiency of both training paradigms. While both paradigms lead to a reasonable approximation of the policy, the model-based approach is more accurate and considerably reduces the number of PDE solves.

math.OC↗

arXiv2024

Differential Equations for Continuous-Time Deep Learning

This short, self-contained article seeks to introduce and survey continuous-time deep learning approaches that are based on neural ordinary differential equations (neural ODEs). It primarily targets readers familiar with ordinary and partial differential equations and their analysis who are curious to see their role in machine learning. Using three examples from machine learning and applied mathematics, we will see how neural ODEs can provide new insights into deep learning and a foundation for more efficient algorithms.

cs.LG↗

arXiv2023

Learning Control Policies of Hodgkin-Huxley Neuronal Dynamics

We present a neural network approach for closed-loop deep brain stimulation (DBS). We cast the problem of finding an optimal neurostimulation strategy as a control problem. In this setting, control policies aim to optimize therapeutic outcomes by tailoring the parameters of a DBS system, typically via electrical stimulation, in real time based on the patient's ongoing neuronal activity. We approximate the value function offline using a neural network to enable generating controls (stimuli) in real time via the feedback form. The neuronal activity is characterized by a nonlinear, stiff system of differential equations as dictated by the Hodgkin-Huxley model. Our training process leverages the relationship between Pontryagin's maximum principle and Hamilton-Jacobi-Bellman equations to update the value function estimates simultaneously. Our numerical experiments illustrate the accuracy of our approach for out-of-distribution samples and the robustness to moderate shocks and disturbances in the system.

math.OC↗

arXiv2023

Efficient Neural Network Approaches for Conditional Optimal Transport with Applications in Bayesian Inference

We present two neural network approaches that approximate the solutions of static and dynamic $\unicode{x1D450}\unicode{x1D45C}\unicode{x1D45B}\unicode{x1D451}\unicode{x1D456}\unicode{x1D461}\unicode{x1D456}\unicode{x1D45C}\unicode{x1D45B}\unicode{x1D44E}\unicode{x1D459}\unicode{x0020}\unicode{x1D45C}\unicode{x1D45D}\unicode{x1D461}\unicode{x1D456}\unicode{x1D45A}\unicode{x1D44E}\unicode{x1D459}\unicode{x0020}\unicode{x1D461}\unicode{x1D45F}\unicode{x1D44E}\unicode{x1D45B}\unicode{x1D460}\unicode{x1D45D}\unicode{x1D45C}\unicode{x1D45F}\unicode{x1D461}$ (COT) problems. Both approaches enable conditional sampling and conditional density estimation, which are core tasks in Bayesian inference$\unicode{x2013}$particularly in the simulation-based ($\unicode{x201C}$likelihood-free$\unicode{x201D}$) setting. Our methods represent the target conditional distribution as a transformation of a tractable reference distribution. Obtaining such a transformation, chosen here to be an approximation of the COT map, is computationally challenging even in moderate dimensions. To improve scalability, our numerical algorithms use neural networks to parameterize candidate maps and further exploit the structure of the COT problem. Our static approach approximates the map as the gradient of a partially input-convex neural network. It uses a novel numerical implementation to increase computational efficiency compared to state-of-the-art alternatives. Our dynamic approach approximates the conditional optimal transport via the flow map of a regularized neural ODE; compared to the static approach, it is slower to train but offers more modeling choices and can lead to faster sampling. We demonstrate both algorithms numerically, comparing them with competing state-of-the-art approaches, using benchmark datasets and simulation-based Bayesian inverse problems.

stat.ML↗

arXiv2023

LSEMINK: A Modified Newton-Krylov Method for Log-Sum-Exp Minimization

This paper introduces LSEMINK, an effective modified Newton-Krylov algorithm geared toward minimizing the log-sum-exp function for a linear model. Problems of this kind arise commonly, for example, in geometric programming and multinomial logistic regression. Although the log-sum-exp function is smooth and convex, standard line search Newton-type methods can become inefficient because the quadratic approximation of the objective function can be unbounded from below. To circumvent this, LSEMINK modifies the Hessian by adding a shift in the row space of the linear model. We show that the shift renders the quadratic approximation to be bounded from below and that the overall scheme converges to a global minimizer under mild assumptions. Our convergence proof also shows that all iterates are in the row space of the linear model, which can be attractive when the model parameters do not have an intuitive meaning, as is common in machine learning. Since LSEMINK uses a Krylov subspace method to compute the search direction, it only requires matrix-vector products with the linear model, which is critical for large-scale problems. Our numerical experiments on image classification and geometric programming illustrate that LSEMINK considerably reduces the time-to-solution and increases the scalability compared to geometric programming and natural gradient descent approaches. It has significantly faster initial convergence than standard Newton-Krylov methods, which is particularly attractive in applications like machine learning. In addition, LSEMINK is more robust to ill-conditioning arising from the nonsmoothness of the problem. We share our MATLAB implementation at https://github.com/KelvinKan/LSEMINK.

math.OC↗

arXiv2023

Alternating Minimization for Regression with Tropical Rational Functions

We propose an alternating minimization heuristic for regression over the space of tropical rational functions with fixed exponents. The method alternates between fitting the numerator and denominator terms via tropical polynomial regression, which is known to admit a closed form solution. We demonstrate the behavior of the alternating minimization method experimentally. Experiments demonstrate that the heuristic provides a reasonable approximation of the input data. Our work is motivated by applications to ReLU neural networks, a popular class of network architectures in the machine learning community which are closely related to tropical rational functions.

math.OC↗

arXiv2023

Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generation

Score-based diffusion models (SBDM) have recently emerged as state-of-the-art approaches for image generation. Existing SBDMs are typically formulated in a finite-dimensional setting, where images are considered as tensors of finite size. This paper develops SBDMs in the infinite-dimensional setting, that is, we model the training data as functions supported on a rectangular domain. In addition to the quest for generating images at ever-higher resolutions, our primary motivation is to create a well-posed infinite-dimensional learning problem that we can discretize consistently on multiple resolution levels. We thereby intend to obtain diffusion models that generalize across different resolution levels and improve the efficiency of the training process. We demonstrate how to overcome two shortcomings of current SBDM approaches in the infinite-dimensional setting. First, we modify the forward process using trace class operators to ensure that the latent distribution is well-defined in the infinite-dimensional setting and derive the reverse processes for finite-dimensional approximations. Second, we illustrate that approximating the score function with an operator network is beneficial for multilevel training. After deriving the convergence of the discretization and the approximation of multilevel training, we demonstrate some practical benefits of our infinite-dimensional SBDM approach on a synthetic Gaussian mixture example, the MNIST dataset, and a dataset generated from a nonlinear 2D reaction-diffusion equation.

cs.LG↗

arXiv2022

Improving Graph Neural Networks with Learnable Propagation Operators

Graph Neural Networks (GNNs) are limited in their propagation operators. In many cases, these operators often contain non-negative elements only and are shared across channels, limiting the expressiveness of GNNs. Moreover, some GNNs suffer from over-smoothing, limiting their depth. On the other hand, Convolutional Neural Networks (CNNs) can learn diverse propagation filters, and phenomena like over-smoothing are typically not apparent in CNNs. In this paper, we bridge these gaps by incorporating trainable channel-wise weighting factors $\omega$ to learn and mix multiple smoothing and sharpening propagation operators at each layer. Our generic method is called $\omega$GNN, and is easy to implement. We study two variants: $\omega$GCN and $\omega$GAT. For $\omega$GCN, we theoretically analyse its behaviour and the impact of $\omega$ on the obtained node features. Our experiments confirm these findings, demonstrating and explaining how both variants do not over-smooth. Additionally, we experiment with 15 real-world datasets on node- and graph-classification tasks, where our $\omega$GCN and $\omega$GAT perform on par with state-of-the-art methods.

cs.LG↗

arXiv2022

A Neural Network Approach for Stochastic Optimal Control

We present a neural network approach for approximating the value function of high-dimensional stochastic control problems. Our training process simultaneously updates our value function estimate and identifies the part of the state space likely to be visited by optimal trajectories. Our approach leverages insights from optimal control theory and the fundamental relation between semi-linear parabolic partial differential equations and forward-backward stochastic differential equations. To focus the sampling on relevant states during neural network training, we use the stochastic Pontryagin maximum principle (PMP) to obtain the optimal controls for the current value function estimate. By design, our approach coincides with the method of characteristics for the non-viscous Hamilton-Jacobi-Bellman equation arising in deterministic control problems. Our training loss consists of a weighted sum of the objective functional of the control problem and penalty terms that enforce the HJB equations along the sampled trajectories. Importantly, training is unsupervised in that it does not require solutions of the control problem. Our numerical experiments highlight our scheme's ability to identify the relevant parts of the state space and produce meaningful value estimates. Using a two-dimensional model problem, we demonstrate the importance of the stochastic PMP to inform the sampling and compare to a finite element approach. With a nonlinear control affine quadcopter example, we illustrate that our approach can handle complicated dynamics. For a 100-dimensional benchmark problem, we demonstrate that our approach improves accuracy and time-to-solution and, via a modification, we show the wider applicability of our scheme.

math.OC↗

arXiv2022

Multivariate Quantile Function Forecaster

We propose Multivariate Quantile Function Forecaster (MQF$^2$), a global probabilistic forecasting method constructed using a multivariate quantile function and investigate its application to multi-horizon forecasting. Prior approaches are either autoregressive, implicitly capturing the dependency structure across time but exhibiting error accumulation with increasing forecast horizons, or multi-horizon sequence-to-sequence models, which do not exhibit error accumulation, but also do typically not model the dependency structure across time steps. MQF$^2$ combines the benefits of both approaches, by directly making predictions in the form of a multivariate quantile function, defined as the gradient of a convex function which we parametrize using input-convex neural networks. By design, the quantile function is monotone with respect to the input quantile levels and hence avoids quantile crossing. We provide two options to train MQF$^2$: with energy score or with maximum likelihood. Experimental results on real-world and synthetic datasets show that our model has comparable performance with state-of-the-art methods in terms of single time step metrics while capturing the time dependency structure.

cs.LG↗

arXiv2021

slimTrain -- A Stochastic Approximation Method for Training Separable Deep Neural Networks

Deep neural networks (DNNs) have shown their success as high-dimensional function approximators in many applications; however, training DNNs can be challenging in general. DNN training is commonly phrased as a stochastic optimization problem whose challenges include non-convexity, non-smoothness, insufficient regularization, and complicated data distributions. Hence, the performance of DNNs on a given task depends crucially on tuning hyperparameters, especially learning rates and regularization parameters. In the absence of theoretical guidelines or prior experience on similar tasks, this requires solving many training problems, which can be time-consuming and demanding on computational resources. This can limit the applicability of DNNs to problems with non-standard, complex, and scarce datasets, e.g., those arising in many scientific applications. To remedy the challenges of DNN training, we propose slimTrain, a stochastic optimization method for training DNNs with reduced sensitivity to the choice hyperparameters and fast initial convergence. The central idea of slimTrain is to exploit the separability inherent in many DNN architectures; that is, we separate the DNN into a nonlinear feature extractor followed by a linear model. This separability allows us to leverage recent advances made for solving large-scale, linear, ill-posed inverse problems. Crucially, for the linear weights, slimTrain does not require a learning rate and automatically adapts the regularization parameter. Since our method operates on mini-batches, its computational overhead per iteration is modest. In our numerical experiments, slimTrain outperforms existing DNN training methods with the recommended hyperparameter settings and reduces the sensitivity of DNN training to the remaining hyperparameters.

cs.LG↗