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Tag: dynamics

  • Neuron Block Dynamics for XOR Classification with Zero-Margin

    Neuron Block Dynamics for XOR Classification with Zero-Margin arXiv:2602.00172v1 Announce Type: new Abstract: The ability of neural networks to learn useful features through stochastic gradient descent (SGD) is a cornerstone of their success. Most theoretical analyses focus on regression or on classification tasks with a positive margin, where worst-case gradient bounds suffice. In contrast, we…

  • The Impact of Anisotropic Covariance Structure on the Training Dynamics and Generalization Error of Linear Networks

    The Impact of Anisotropic Covariance Structure on the Training Dynamics and Generalization Error of Linear Networks arXiv:2601.06961v1 Announce Type: new Abstract: The success of deep neural networks largely depends on the statistical structure of the training data. While learning dynamics and generalization on isotropic data are well-established, the impact of pronounced anisotropy on these crucial…

  • Exact Dynamics of Multi-class Stochastic Gradient Descent

    Exact Dynamics of Multi-class Stochastic Gradient Descent arXiv:2510.14074v1 Announce Type: new Abstract: We develop a framework for analyzing the training and learning rate dynamics on a variety of high- dimensional optimization problems trained using one-pass stochastic gradient descent (SGD) with data generated from multiple anisotropic classes. We give exact expressions for a large class of…

  • Neural Stochastic Differential Equations on Compact State-Spaces

    Neural Stochastic Differential Equations on Compact State-Spaces arXiv:2508.17090v1 Announce Type: new Abstract: Many modern probabilistic models rely on SDEs, but their adoption is hampered by instability, poor inductive bias outside bounded domains, and reliance on restrictive dynamics or training tricks. While recent work constrains SDEs to compact spaces using reflected dynamics, these approaches lack continuous…

  • Learning quadratic neural networks in high dimensions: SGD dynamics and scaling laws

    Learning quadratic neural networks in high dimensions: SGD dynamics and scaling laws arXiv:2508.03688v1 Announce Type: new Abstract: We study the optimization and sample complexity of gradient-based training of a two-layer neural network with quadratic activation function in the high-dimensional regime, where the data is generated as $y propto sum_{j=1}^{r}lambda_j sigmaleft(langle boldsymbol{theta_j}, boldsymbol{x}rangleright), boldsymbol{x} sim N(0,boldsymbol{I}_d)$,…

  • Stochastic forest transition model dynamics and parameter estimation via deep learning

    Stochastic forest transition model dynamics and parameter estimation via deep learning arXiv:2507.21486v1 Announce Type: new Abstract: Forest transitions, characterized by dynamic shifts between forest, agricultural, and abandoned lands, are complex phenomena. This study developed a stochastic differential equation model to capture the intricate dynamics of these transitions. We established the existence of global positive solutions…

  • Critically-Damped Higher-Order Langevin Dynamics

    Critically-Damped Higher-Order Langevin Dynamics arXiv:2506.21741v1 Announce Type: new Abstract: Denoising Diffusion Probabilistic Models represent an entirely new class of generative AI methods that have yet to be fully explored. Critical damping has been successfully introduced in Critically-Damped Langevin Dynamics (CLD) and Critically-Damped Third-Order Langevin Dynamics (TOLD++), but has not yet been applied to dynamics of…

  • Oh SnapMMD! Forecasting Stochastic Dynamics Beyond the Schr”odinger Bridge’s End

    Oh SnapMMD! Forecasting Stochastic Dynamics Beyond the Schr”odinger Bridge’s End arXiv:2505.16082v1 Announce Type: new Abstract: Scientists often want to make predictions beyond the observed time horizon of “snapshot” data following latent stochastic dynamics. For example, in time course single-cell mRNA profiling, scientists have access to cellular transcriptional state measurements (snapshots) from different biological replicates at…

  • Propagation of Chaos in One-hidden-layer Neural Networks beyond Logarithmic Time

    Propagation of Chaos in One-hidden-layer Neural Networks beyond Logarithmic Time arXiv:2504.13110v1 Announce Type: new Abstract: We study the approximation gap between the dynamics of a polynomial-width neural network and its infinite-width counterpart, both trained using projected gradient descent in the mean-field scaling regime. We demonstrate how to tightly bound this approximation gap through a differential…

  • A Constant Velocity Latent Dynamics Approach for Accelerating Simulation of Stiff Nonlinear Systems

    A Constant Velocity Latent Dynamics Approach for Accelerating Simulation of Stiff Nonlinear Systems arXiv:2501.08423v1 Announce Type: new Abstract: Solving stiff ordinary differential equations (StODEs) requires sophisticated numerical solvers, which are often computationally expensive. In particular, StODE’s often cannot be solved with traditional explicit time integration schemes and one must resort to costly implicit methods to…

  • Time-Reversible Bridges of Data with Machine Learning

    Time-Reversible Bridges of Data with Machine Learning arXiv:2412.13665v1 Announce Type: new Abstract: The analysis of dynamical systems is a fundamental tool in the natural sciences and engineering. It is used to understand the evolution of systems as large as entire galaxies and as small as individual molecules. With predefined conditions on the evolution of dy-namical…