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Interpretable reduced-order modeling with time-scale separation
Machine Learning · Submitted 3 Mar 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We propose such a data-driven scheme that automates the identification of the time-scales involved and can produce stable predictions forward in time as well as under different initial conditions not included in the training data.
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The abstract, sentence by sentence
Partial Differential Equations (PDEs) with high dimensionality are commonly encountered in computational physics and engineering. However, finding solutions for these PDEs can be computationally expensive, making model-order reduction crucial. We propose such a data-driven scheme that automates the identification of the time-scales involved and can produce stable predictions forward in time as well as under different initial conditions not included in the training data. To this end, we combine a non-linear autoencoder architecture with a time-continuous model for the latent dynamics in the complex space. It readily allows for the inclusion of sparse and irregularly sampled training data. The learned, latent dynamics are interpretable and reveal the different temporal scales involved. We show that this data-driven scheme can automatically learn the independent processes that decompose a system of linear ODEs along the eigenvectors of the system's matrix. Apart from this, we demonstrate the applicability of the proposed framework in a hidden Markov Model and the (discretized) Kuramoto-Shivashinsky (KS) equation. Additionally, we propose a probabilistic version, which captures predictive uncertainties and further improves upon the results of the deterministic framework.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Partial Differential Equations (PDEs) with high dimensionality are commonly encountered in computational physics and engineering. | background 1.00 | 0.04 | 0.02 | asserted 0.99 |
| s1 | However, finding solutions for these PDEs can be computationally expensive, making model-order reduction crucial. | background 0.95 | 0.05 | 0.03 | asserted 0.99 |
| s2 | We propose such a data-driven scheme that automates the identification of the time-scales involved and can produce stable predictions forward in time as well as under different initial conditions not included in the training data. | contribution 0.98 | 0.89 | 0.42 | asserted 0.95 |
| s3 | To this end, we combine a non-linear autoencoder architecture with a time-continuous model for the latent dynamics in the complex space. | method 0.99 | 0.88 | 0.05 | asserted 0.99 |
| s4 | It readily allows for the inclusion of sparse and irregularly sampled training data. | method 0.85 | 0.68 | 0.03 | asserted 0.97 |
| s5 | The learned, latent dynamics are interpretable and reveal the different temporal scales involved. | result 0.61 | 0.90 | 0.19 | asserted 1.00 |
| s6 | We show that this data-driven scheme can automatically learn the independent processes that decompose a system of linear ODEs along the eigenvectors of the system's matrix. | result 0.97 | 0.94 | 0.18 | asserted 1.00 |
| s7 | Apart from this, we demonstrate the applicability of the proposed framework in a hidden Markov Model and the (discretized) Kuramoto-Shivashinsky (KS) equation. | result 0.50 | 0.94 | 0.10 | asserted 1.00 |
| s8 | Additionally, we propose a probabilistic version, which captures predictive uncertainties and further improves upon the results of the deterministic framework. | contribution 0.95 | 0.94 | 0.14 | asserted 1.00 |
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The paper
Original abstract
Partial Differential Equations (PDEs) with high dimensionality are commonly encountered in computational physics and engineering. However, finding solutions for these PDEs can be computationally expensive, making model-order reduction crucial. We propose such a data-driven scheme that automates the identification of the time-scales involved and can produce stable predictions forward in time as well as under different initial conditions not included in the training data. To this end, we combine a non-linear autoencoder architecture with a time-continuous model for the latent dynamics in the complex space. It readily allows for the inclusion of sparse and irregularly sampled training data. The learned, latent dynamics are interpretable and reveal the different temporal scales involved. We show that this data-driven scheme can automatically learn the independent processes that decompose a system of linear ODEs along the eigenvectors of the system's matrix. Apart from this, we demonstrate the applicability of the proposed framework in a hidden Markov Model and the (discretized) Kuramoto-Shivashinsky (KS) equation. Additionally, we propose a probabilistic version, which captures predictive uncertainties and further improves upon the results of the deterministic framework.
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How the literature uses it
3 arXiv papers cite this in 4 sentences
Papers citing it · 3 sentences without a confident label
Contests it 1 paper
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Similarly, linear decomposition techniques such as Dynamic Mode Decomposition (DMD) [ref] or Proper Orthogonal Decomposition (POD) [ref] , as well as interpretable reduced-order models that impose linear dynamics [ref] , often fall short in capturing nonlinear behavior.
Automated Discovery of Operable Dynamics from Videos · Introduction · contest 0.83
What it cites · 1 sentence without a confident label
Uses the method or tool of 3 papers
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We are also proposing a probabilistic version of our algorithm [ref] that makes use of probabilistic Slow Feature Analysis [ref] .
Physics-aware, probabilistic model order reduction with guaranteed stability · Introduction · 0.88
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We compute Monte Carlo estimates of the gradient of the ELBO with respect to $$ and $$ with the help of the reparametrization trick [ref] and carry out stochastic optimization with the ADAM algorithm [ref] .
Auto-Encoding Variational Bayes · Inference and Learning · 1.00
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We compute Monte Carlo estimates of the gradient of the ELBO with respect to $$ and $$ with the help of the reparametrization trick [ref] and carry out stochastic optimization with the ADAM algorithm [ref] .
Adam: A Method for Stochastic Optimization · Inference and Learning · 1.00
Cites as support 1 paper
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The deployment of interpretable latent dynamics ensures that conservation of important properties of that are reflected in the reduced-order model [ref] .
Physics-aware, probabilistic model order reduction with guaranteed stability · Introduction · 0.98
Cites as background 1 paper
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Apart from using an architecture based on autoencoders to identify the latent space, projection-based methods could also be employed [ref] .
Reduced Order Model Closures: A Brief Tutorial · Introduction · 0.85
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