DeepONet learns operators for PDEs with varying parameters and initial conditions.
problem Learning operators for partial differential equations with different parameters or initial conditions.
method DeepONet uses a Branch net and Trunk net to minimize error between evaluated and expected outputs, incorporating a scalar auxiliary variable approach for energy dissipation.
result DeepONet can accurately approximate operators for PDEs with varying parameters or initial conditions.
Generic level sets in mean curvature flow are BV solutions.
problem Understanding the behavior of level sets in mean curvature flow.
method Using the framework of sets of finite perimeter and distributional solutions, the paper extends Evans and Spruck's work.
result Generic level sets are distributional solutions with optimal energy dissipation rate.
VAV method optimizes learning rate for faster, stable SGD convergence.
problem Optimizing learning rate for efficient and stable machine learning models.
method Energy-based self-adaptive learning rate with auxiliary variable r. result VAV method achieves faster convergence and superior stability with larger learning rates.
A new optimization algorithm improves convergence in unconstrained problems.
problem Unconstrained optimization problems.
method Element-wise relaxed scalar auxiliary variable (E-RSAV) algorithm.
result Improved convergence and alignment of modified and original energy.
Transfer learning improves chaotic dynamics predictions with less data.
problem Efficiently predicting chaotic dynamics with limited data.
method Transfer learning for nonlinear dynamics, optimizing transfer rate and leveraging small-scale turbulence universality.
result Significantly more accurate inference of chaotic dynamics achieved.
Bayesian inference over admissible histories leads to irreversible kinetics.
problem Modeling irreversible processes in systems with uncertain histories.
method A Gibbs-type measure weighted by energy-dissipation action and observation constraints, interpreted as a Bayesian posterior.
result The measure concentrates on maximum-a-posteriori (MAP) histories, recovering classical deterministic evolution.
Paper adds Fisher Information to mean field optimization for faster convergence.
problem Mean field optimization in neural networks training.
method Developed energy-dissipation method and gradient flow on probability space.
result Marginal distributions converge exponentially to minimizer.
A new EVI framework improves ParVI methods by maintaining variational structure and reducing KL-divergence.
problem Improving variational inference methods for better approximation of target distributions.
method EVI framework that minimizes the VI objective function based on an energy-dissipation law, including a new 'Approximation-then-Variation' scheme.
result The new scheme significantly decreases KL-divergence and outperforms existing ParVI methods in fidelity.
The last decade has shown a tremendous success in solving various computer vision problems with the help of deep learning techniques. Lately, many works have demonstrated that learning-based approaches with suitable network architectures even exhibit superior performance for the solution of (ill-posed) image reconstruc…
D-LinOSS models learn to dissipate energy, improving performance on long-range tasks.
problem Representational limitations of LinOSS models in long-range reasoning.
method Introducing Damped Linear Oscillatory State-Space models (D-LinOSS) that learn to dissipate latent state energy on arbitrary time scales.
result D-LinOSS consistently outperforms previous LinOSS methods on long-range learning tasks, achieving faster convergence and reducing hyperparameter search space.
In this work, we introduce Dissipative SymODEN, a deep learning architecture which can infer the dynamics of a physical system with dissipation from observed state trajectories. To improve prediction accuracy while reducing network size, Dissipative SymODEN encodes the port-Hamiltonian dynamics with energy dissipation …
FRAME (Filters, Random fields, And Maximum Entropy) is an energy-based descriptive model that synthesizes visual realism by capturing mutual patterns from structural input signals. The maximum likelihood estimation (MLE) is applied by default, yet conventionally causes the unstable training energy that wrecks the gener…
A new shape space allows optimization of non-smooth shapes in fluid mechanics.
problem Optimizing non-smooth shapes in fluid mechanics.
method Constructing a product manifold to include piecewise-smooth shapes.
result Numerical results show applicability in minimizing viscous energy dissipation.
The article reviews how gradient flow systems on hypergraphs connect to information geometry and nonequilibrium physics.
problem Understanding the geometry of perturbed gradient flow systems on hypergraphs.
method Formulating modern nonequilibrium principles within the framework of perturbed gradient flow systems on hypergraphs.
result New concepts like moduli spaces and thermodynamical area are introduced to understand speed limits.
Paper explores Fisher-Rao gradient flows and their kernel approximations.
problem Understanding and analyzing approximations of Fisher-Rao gradient flows.
method Rigorous investigation of Fisher-Rao and Wasserstein type gradient flows, focusing on kernel approximations.
result Proves evolutionary Γ-convergence for kernel-approximated Fisher-Rao flows, providing theoretical guarantees.
Major advancements in building general-purpose and customized hardware have been one of the key enablers of versatility and pervasiveness of machine learning models such as deep neural networks. To sustain this ubiquitous deployment of machine learning models and cope with their computational and storage complexity, se…
Develops a mean-field theory for multi-head self-attention under cross-entropy training.
problem Mean-field analysis of multi-head self-attention under cross-entropy training.
method Mean-field theory for a simplified single-layer causal multi-head self-attention model.
result Proves a static finite-head approximation bound for the optimal risk.
JKO scheme adds deceleration in rapidly changing metric curvature directions.
problem Understanding the implicit bias of the JKO scheme in Wasserstein gradient flow.
method Characterized the implicit bias of the JKO scheme at second order in η, modifying the energy functional.
result JKO scheme adds deceleration in directions where metric curvature of J is rapidly changing.