mlsanityData to 7 Oct 2026

On latent dynamics learning in nonlinear reduced order modeling

arXiv 2408.15183 ↗First posted 27 Aug 2024math.NA cs.LG cs.NAcited by 14

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Abstract (arXiv)

In this work, we present the novel mathematical framework of latent dynamics models (LDMs) for reduced order modeling of parameterized nonlinear time-dependent PDEs. Our framework casts this latter task as a nonlinear dimensionality reduction problem, while constraining the latent state to evolve accordingly to an (unknown) dynamical system. A time-continuous setting is employed to derive error and stability estimates for the LDM approximation of the full order model (FOM) solution. We analyze the impact of using an explicit Runge-Kutta scheme in the time-discrete setting, resulting in the DeltaLDM formulation, and further explore the learnable setting, DeltaLDMθ, where deep neural networks approximate the discrete LDM components, while providing a bounded approximation error with respect to the FOM. Moreover, we extend the concept of parameterized Neural ODE - recently proposed as a possible way to build data-driven dynamical systems with varying input parameters - to be a convolutional architecture, where the input parameters information is injected by means of an affine modulation mechanism, while designing a convolutional autoencoder neural network able to retain spatial-coherence, thus enhancing interpretability at the latent level. Numerical experiments, including the Burgers' and the advection-reaction-diffusion equations, demonstrate the framework's ability to obtain, in a multi-query context, a time-continuous approximation of the FOM solution, thus being able to query the LDM approximation at any given time instance while retaining a prescribed level of accuracy. Our findings highlight the remarkable potential of the proposed LDMs, representing a mathematically rigorous framework to enhance the accuracy and approximation capabilities of reduced order modeling for time-dependent parameterized PDEs.

Key facts. Cited by 14 papers in the map (data through 7 Oct 2026). Builds on 50 of its 113 references that are in the map. First posted on arXiv 27 Aug 2024.

Builds on

50 of 113 references.

References · 50

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38 more
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 6; 2026: 8.

14 papers.

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  1. 2025Reduced Order Modeling with Shallow Recurrent Decoder Networkscited by 24
  2. 2025Sparse identification of nonlinear dynamics and Koopman operators with Shallow Recurrent Decoder Networkscited by 19
  3. 2025RONOM: Reduced-Order Neural Operator Modelingcited by 2
  4. 2025Neural non-canonical Hamiltonian dynamics for long-time simulationscited by 2
  5. 2026On Geometry Regularization in Autoencoder Reduced-Order Models with Latent Neural ODE Dynamicscited by 1
  6. 2026A deep learning framework for jointly solving transient Fokker-Planck equations with arbitrary parameters and initial distributionscited by 1
  7. 2025Beyond Static Models: Hypernetworks for Adaptive and Generalizable Forecasting in Complex Parametric Dynamical Systemscited by 0
  8. 2025Revealing Low-Dimensional Structure in 2D Richtmyer-Meshkov Instabilities via Parametric Reduced-Order Modelingcited by 0
  9. 2026Latent Dynamics Graph Convolutional Networks for model order reduction of parameterized time-dependent PDEscited by 0
  10. 2026AE-ViT: Stable Long-Horizon Parametric Partial Differential Equations Modelingcited by 0
  11. 2026A Quadratic Order Reduction -- Gaussian Process Ordinary Differential Equation framework for the inference of Large Continuous Dynamical Systemscited by 0
  12. 2026Convolutional Symmetric AutoEncoders: enhancing latent stability via differential geometrycited by 0
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  1. 2026No Free Lunch in Flow Surrogates under Time-Varying Boundary Conditions: A Two-Regime Studycited by 0
  2. 2026Evaluating Hybrid Quantum-Classical Models for Reduced-Order Brain Deformation Dynamicscited by 0