[Paper Review] Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutions

Paolo Conti, Giorgio Gobat+3 more|arXiv (Cornell University)|Nov 13, 2022

Model Reduction and Neural Networks4 citations

TL;DR

This paper presents a non-intrusive, data-driven reduced order modeling framework that combines autoencoders (AEs) and sparse identification of nonlinear dynamics (SINDy) to construct low-dimensional dynamical systems from limited full-order simulation data. By embedding parametric dependencies into the identified latent ODEs, the method enables efficient continuation of periodic solutions and accurate prediction of steady-state responses across time and parameter ranges, even beyond training data, with demonstrated success in structural and fluid dynamics problems.

ABSTRACT

Highly accurate simulations of complex phenomena governed by partial differential equations (PDEs) typically require intrusive methods and entail expensive computational costs, which might become prohibitive when approximating steady-state solutions of PDEs for multiple combinations of control parameters and initial conditions. Therefore, constructing efficient reduced order models (ROMs) that enable accurate but fast predictions, while retaining the dynamical characteristics of the physical phenomenon as parameters vary, is of paramount importance. In this work, a data-driven, non-intrusive framework which combines ROM construction with reduced dynamics identification, is presented. Starting from a limited amount of full order solutions, the proposed approach leverages autoencoder neural networks with parametric sparse identification of nonlinear dynamics (SINDy) to construct a low-dimensional dynamical model. This model can be queried to efficiently compute full-time solutions at new parameter instances, as well as directly fed to continuation algorithms. These aim at tracking the evolution of periodic steady-state responses as functions of system parameters, avoiding the computation of the transient phase, and allowing to detect instabilities and bifurcations. Featuring an explicit and parametrized modeling of the reduced dynamics, the proposed data-driven framework presents remarkable capabilities to generalize with respect to both time and parameters. Applications to structural mechanics and fluid dynamics problems illustrate the effectiveness and accuracy of the proposed method.

Motivation & Objective

• Address the high computational cost of full-order models (FOMs) for parametrized, time-dependent PDEs across multiple parameter and initial condition combinations.
• Overcome limitations of intrusive ROMs and standard data-driven surrogates in interpretability, generalization, and stability under temporal or parametric extrapolation.
• Develop a non-intrusive, interpretable, and generalizable reduced order model that captures the intrinsic dynamics of complex systems via latent-space system identification.
• Enable continuation of periodic steady-state solutions without simulating transient phases, facilitating bifurcation detection and instability analysis.
• Construct a framework that generalizes beyond training data in both time and parameter space, leveraging physical consistency through SINDy.

Proposed method

• Use proper orthogonal decomposition (POD) to extract spatial modes from full-order solution snapshots.
• Train a feedforward autoencoder to map high-dimensional solution fields to a low-dimensional latent space, preserving essential dynamics.
• Apply parametric sparse identification of nonlinear dynamics (SINDy) to identify a sparse system of ordinary differential equations (ODEs) in the latent space, incorporating parameter dependence.
• Construct a parametrized latent ODE system that explicitly models how dynamics evolve with system parameters.
• Use the identified latent ODE model for long-time integration and numerical continuation to track periodic solutions and bifurcations.
• Reconstruct full-order solutions via the decoder and POD modes, enabling physical interpretation and validation.

Experimental results

Research questions

• RQ1 Can a data-driven, non-intrusive ROM framework combine autoencoders and SINDy to accurately model the dynamics of parametrized PDEs with minimal training data?
• RQ2 To what extent can the identified latent dynamical system generalize to new parameter values and long-time horizons beyond the training data range?
• RQ3 Can the parametric latent ODE model support continuation techniques to efficiently compute periodic steady-state solutions without simulating transients?
• RQ4 How well does the framework detect bifurcations and transitions between flow regimes (e.g., laminar to unsteady flow) in fluid dynamics problems?
• RQ5 Does incorporating SINDy-based sparsity and physical consistency improve robustness and reduce overfitting compared to standard deep learning-based ROMs?

Key findings

• The AE+SINDy framework successfully identifies a one-dimensional latent dynamical system for a clamped-clamped beam, enabling accurate prediction of steady-state responses for new parameters outside the training range.
• For the fluid flow past a cylinder, the method accurately captures both laminar and unsteady flow regimes and correctly identifies the Reynolds number at which the flow transitions to vortex shedding.
• The bifurcation diagram computed via continuation using the latent model aligns closely with the physical behavior observed in full-order snapshots, confirming the method’s accuracy in detecting qualitative changes in dynamics.
• The latent ODE model enables stable and accurate long-time integration of periodic solutions, even for testing instances far from training data, demonstrating robustness to temporal and parametric extrapolation.
• The framework achieves high accuracy at low computational cost while remaining non-intrusive, requiring only a limited number of FOM snapshots and no access to the underlying PDE operators during inference.
• The integration of SINDy with parametric latent dynamics enhances interpretability and reduces overfitting compared to purely black-box deep learning surrogates, as evidenced by stable continuation and generalization performance.