Learning symplectic model reduction based on a approximation theorem of symplectic embeddings

• URL: http://arxiv.org/abs/2606.04623v1
• Date: Wed, 03 Jun 2026 08:58:44 GMT
• Title: Learning symplectic model reduction based on a approximation theorem of symplectic embeddings
• Abstract summary: High-dimensional Hamiltonian systems play a central role in many scientific and engineering disciplines.<n>Deep learning provides powerful tools for constructing low-dimensional surrogates from data.<n>In intrinsic symplectic structure is easily destroyed during model reduction.
• Score: 3.028436968161472
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: High-dimensional Hamiltonian systems play a central role in many scientific and engineering disciplines, with dynamics evolving on symplectic manifolds. Although deep learning provides powerful tools for constructing low-dimensional surrogates from data, the intrinsic symplectic structure is easily destroyed during model reduction. As a result, a standard autoencoder may produce latent coordinates that do not support a Hamiltonian flow, leading to unstable long-time prediction. In this paper, we first establish a universal approximation theorem for symplectic embeddings. Based on this theory, we propose symplecticity-preserving autoencoders (SpAE), in which the decoder is parameterized as a symplectic embedding and the encoder is constructed as the corresponding symplectic projection. This architecture is expressive enough to approximate nonlinear symplectic embeddings and the associated symplectic projections, preserves the symplectic structure exactly by construction, and can be trained by standard unconstrained optimization, thereby improving both reconstruction and prediction accuracy. Extensive experiments on high-dimensional lattice and particle systems demonstrate the effectiveness of the proposed method.

Related papers

• CoSynFlow: Conformal Symplectic Neural Flows for Cross-System Prediction of Dissipative Hamiltonian Dynamics [8.221727426707526]
  We propose a conformal symplectic neural flow for learning continuous-time solution maps of dissipative Hamiltonian dynamics.<n>CoSynFlow composes symplectic shear maps with explicit conformal scaling, preserving the conformal symplectic structure by construction.
  arXiv  Detail & Related papers  (2026-08-01T10:12:58Z)
• SIGMA: Scalable Spectral Insights for LLM Collapse [51.863164847253366]
  We introduce SIGMA (Spectral Inequalities for Gram Matrix Analysis), a unified framework for model collapse.<n>By utilizing benchmarks that deriving and deterministic bounds on the matrix's spectrum, SIGMA provides a mathematically grounded metric to track the contraction of the representation space.<n>We demonstrate that SIGMA effectively captures the transition towards states, offering both theoretical insights into the mechanics of collapse.
  arXiv  Detail & Related papers  (2026-01-06T19:47:11Z)
• Reduced-order modeling of Hamiltonian dynamics based on symplectic neural networks [9.050817345496709]
  We introduce a novel data-driven symplectic induced-order modeling (ROM) framework for high-dimensional Hamiltonian systems.<n>Rom unifies latent-space discovery and dynamics learning within a single, end-to-end neural architecture.
  arXiv  Detail & Related papers  (2025-08-16T05:09:28Z)
• Efficient Parametric SVD of Koopman Operator for Stochastic Dynamical Systems [51.54065545849027]
  The Koopman operator provides a principled framework for analyzing nonlinear dynamical systems.<n>VAMPnet and DPNet have been proposed to learn the leading singular subspaces of the Koopman operator.<n>We propose a scalable and conceptually simple method for learning the top-$k$ singular functions of the Koopman operator.
  arXiv  Detail & Related papers  (2025-07-09T18:55:48Z)
• Certified Neural Approximations of Nonlinear Dynamics [51.01318247729693]
  In safety-critical contexts, the use of neural approximations requires formal bounds on their closeness to the underlying system.<n>We propose a novel, adaptive, and parallelizable verification method based on certified first-order models.
  arXiv  Detail & Related papers  (2025-05-21T13:22:20Z)
• Generalized Factor Neural Network Model for High-dimensional Regression [50.554377879576066]
  We tackle the challenges of modeling high-dimensional data sets with latent low-dimensional structures hidden within complex, non-linear, and noisy relationships.<n>Our approach enables a seamless integration of concepts from non-parametric regression, factor models, and neural networks for high-dimensional regression.
  arXiv  Detail & Related papers  (2025-02-16T23:13:55Z)
• The Convex Landscape of Neural Networks: Characterizing Global Optima and Stationary Points via Lasso Models [75.33431791218302]
  Deep Neural Network Network (DNN) models are used for programming purposes. In this paper we examine the use of convex neural recovery models. We show that all the stationary non-dimensional objective objective can be characterized as the standard a global subsampled convex solvers program. We also show that all the stationary non-dimensional objective objective can be characterized as the standard a global subsampled convex solvers program.
  arXiv  Detail & Related papers  (2023-12-19T23:04:56Z)
• Symplectic Autoencoders for Model Reduction of Hamiltonian Systems [0.0]
  It is crucial to preserve the symplectic structure associated with the system in order to ensure long-term numerical stability. We propose a new neural network architecture in the spirit of autoencoders, which are established tools for dimension reduction. In order to train the network, a non-standard gradient descent approach is applied.
  arXiv  Detail & Related papers  (2023-12-15T18:20:25Z)
• Deep Learning for Structure-Preserving Universal Stable Koopman-Inspired Embeddings for Nonlinear Canonical Hamiltonian Dynamics [9.599029891108229]
  We focus on the identification of global linearized embeddings for canonical nonlinear Hamiltonian systems through a symplectic transformation. To overcome the shortcomings of Koopman operators for systems with continuous spectra, we apply the lifting principle and learn global cubicized embeddings. We demonstrate the capabilities of deep learning in acquiring compact symplectic coordinate transformation and the corresponding simple dynamical models.
  arXiv  Detail & Related papers  (2023-08-26T09:58:09Z)
• Learning Nonlinear Projections for Reduced-Order Modeling of Dynamical Systems using Constrained Autoencoders [0.0]
  We introduce a class of nonlinear projections described by constrained autoencoder neural networks in which both the manifold and the projection fibers are learned from data. Our architecture uses invertible activation functions and biorthogonal weight matrices to ensure that the encoder is a left inverse of the decoder. We also introduce new dynamics-aware cost functions that promote learning of oblique projection fibers that account for fast dynamics and nonnormality.
  arXiv  Detail & Related papers  (2023-07-28T04:01:48Z)
• Semi-Supervised Manifold Learning with Complexity Decoupled Chart Autoencoders [45.29194877564103]
  This work introduces a chart autoencoder with an asymmetric encoding-decoding process that can incorporate additional semi-supervised information such as class labels. We discuss the approximation power of such networks and derive a bound that essentially depends on the intrinsic dimension of the data manifold rather than the dimension of ambient space.
  arXiv  Detail & Related papers  (2022-08-22T19:58:03Z)
• On dissipative symplectic integration with applications to gradient-based optimization [77.34726150561087]
  We propose a geometric framework in which discretizations can be realized systematically. We show that a generalization of symplectic to nonconservative and in particular dissipative Hamiltonian systems is able to preserve rates of convergence up to a controlled error.
  arXiv  Detail & Related papers  (2020-04-15T00:36:49Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.