Blending data and physics for reduced-order modeling of systems with spatiotemporal chaotic dynamics

summary

The gist

The gist: This work develops a hybrid reduced-order model (ROM), informed by both data and physics, for evolving spatiotemporal chaotic dynamics on an invariant manifold whose coordinates are found

In short

This work develops a hybrid reduced-order model (ROM) called physics-informed DManD to study chaotic dynamics. It combines data-driven methods using autoencoders with physics-based projections from a full order model. The resulting models show superior predictive accuracy and robustness compared to purely data-driven models, even when the underlying physical model has errors.

Key concepts

Hybrid ROM Framework (PI-DManD): This framework merges two methods: Data-driven Manifold Dynamics (DManD) and Manifold Galerkin (MG). It uses a neural ODE to correct errors from the physics projection or uses the physical model as a prior for training the data-driven component, creating a combined, more robust modeling approach.
Data-driven Manifold Dynamics (DManD): This step uses an autoencoder to compress high-dimensional system data into a low-dimensional latent space. The model then learns the dynamics within this compressed space by approximating the true vector field using a neural network, allowing for reduced-order evolution.
Manifold Galerkin (MG): MG projects the physics of a full order model onto the learned low-dimensional manifold coordinates. While it provides a physical basis for the reduced model, it is computationally expensive because it requires decoding latent coordinates back to high dimensions at every step.
Physics-informed DManD (PI-DManD): This involves incorporating physics into the data-driven approach. Specifically, PI-DManD uses a correction term to fix numerical errors or uses the physical model as a statistical prior when training the neural network approximation of the dynamics in latent space.

Terminology used across episodes

Reinforcement Learning (RL) — explained in 25 episodes
Retrieval-Augmented Generation (RAG) — explained in 21 episodes
Sample Complexity — explained in 20 episodes
Flow Matching — explained in 19 episodes
Quantum Fisher Information (QFI) — explained in 19 episodes
Federated Learning — explained in 17 episodes

This episode discusses

The paper

Blending data and physics for reduced-order modeling of systems with spatiotemporal chaotic dynamics · Read on arXiv

Alex Guo, Michael D. Graham

Department of Chemical and Biological Engineering, University of Wisconsin-Madison

While data-driven techniques are powerful tools for reduced-order modeling of systems with chaotic dynamics, great potential remains for leveraging known physics (i.e. a full-order model (FOM)) to improve predictive capability. We develop a hybrid reduced order model (ROM), informed by both data and FOM, for evolving spatiotemporal chaotic dynamics on an invariant manifold whose coordinates are found using an autoencoder. This approach projects the vector field of the FOM onto the invariant manifold; then, this physics-derived vector field is either corrected using dynamic data, or used as a Bayesian prior that is updated with data. In both cases, the neural ordinary differential equation approach is used. We consider simulated data from the Kuramoto-Sivashinsky and complex Ginzburg-Landau equations. Relative to the data-only approach, for scenarios of abundant data, scarce data, and even an incorrect FOM (i.e. erroneous parameter values), the hybrid approach yields substantially improved time-series predictions.

Transcript

Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: We're looking at the paper: "Blending data and physics for reduced-order modeling of systems with spatiotemporal chaotic dynamics".

Jane: Here's the gist of the paper. The gist: This work develops a hybrid reduced-order model (ROM), informed by both data and physics, for evolving spatiotemporal chaotic dynamics on an invariant manifold whose coordinates are found using an autoencoder.

Tom: Hybrid ROM Framework The approach combines the purely data-driven method Data-driven manifold dynamics (DManD) with the physics-based Manifold Galerkin (MG) method to create a hybrid ROM framework called physics-informed DManD (PIDManD) The hybrid ROM is built by combining MG-N with DManD in two ways, which are illustrated in the bottom row of Figure 1(c).

Jane: The first method, PI-DManD-c, adds a correction term learned by a NODE to the MG-N vector field in order to account for numerical and inherent errors in the FOM.

Tom: The second hybrid ROM is PI-DManD-B, which uses the MG-N model as a Bayesian prior for training a NODE model.

Title and authors: Jane: So what’s the actual summary of this "Blending data and physics for reduced-order modeling of systems with spatiotemporal chaotic dynamics" paper? What exactly are they proposing to do?

Tom: Well, it outlines a hybrid approach called physics-informed DManD, which combines data-driven manifold dynamics with the Manifold Galerkin method. They use an autoencoder first to find the low-dimensional latent coordinates of the system.

Lu: The autoencoder part maps high-dimensional data onto a lower dimensional space, say h, where d h is smaller than the original dimension, and they assume this latent space h represents where the system’s behavior actually lies, which is an invariant manifold.

Meng: So they're using that latent representation to learn a new dynamical system in that reduced space, dh over dt = g(h), which they approximate with a neural network. That sounds like standard data-driven ROM setup.

Lalam: But the hybrid part is where it gets interesting because they integrate the physics of the full order model, or FOM, into that process to make it better than just using data alone.

Tom: Right, so they take that physics-derived vector field and either correct it using some learned term from a neural ordinary differential equation approach, or they use the physics model as a prior when training another neural ODE model.

The paper's summary: Jane: So, if I had to explain this to someone who doesn't read papers, what is the main takeaway from their summary? What’s the big picture of how they are blending data and physics?

Lu: The main point is that instead of relying only on data to discover the low-dimensional space where the chaos lives, they use physics knowledge—the full order model—to guide or correct that discovery.

Tom: They project the vector field from the FOM onto this manifold they found with the autoencoder, and then they either add a correction term learned by a neural ordinary differential equation to that vector field, or they use it as a Bayesian prior for training another neural ODE model.

Meng: I see them using two specific hybrid methods: PI-DManD-c adds a correction term g c(h) to the MG-N vector field, and PI-DManD-B uses the MG-N model as a Bayesian prior. That sounds like they have two ways to inject physics knowledge into the data part of things.

Lalam: The key is that in both cases, they are using neural ordinary differential equations to define how this system evolves in the latent coordinates, which is what allows them to get the dynamics running.

The paper's improvements: Jane: Okay, so they say this hybrid approach is better than just a data-only method? What are the specific improvements they claim for things like prediction accuracy?

Tom: They show substantial improvements in predictive capability compared to an approach using only data. For scenarios with abundant data and an exact FOM, both PI-DManD-B and PI-DManD-c accurately track the true solution for much longer periods.

Lu: They report that for scenarios of abundant data, scarce data, and even an incorrect FOM—meaning when the physics part isn't perfectly right—the hybrid approach still yields substantially improved time-series predictions over just using the data.

Meng: That’s important because in production, we never have perfect physics models. The fact that PI-DManD-B still performs well when you use an erroneous parameter value, like L=twenty in the FOM, shows it's more robust than pure data methods <ref:2507.21299#pg1>.

Lalam: They show that for abundant data and an exact FOM, all five methods they tested performed similarly when looking at long-time statistics, using the Earth mover’s distance between model predictions and ground truth trajectories.

Conclusion: Tom: So to wrap up this paper, the big conclusion is that including physics-based prior information substantially improves the predictive capabilities of data-driven ROMs for systems with spatiotemporal chaotic dynamics, even when that physics information isn't completely correct.

Jane: That’s a strong statement because it means we don't need a perfect physical model to get good predictions from data alone if we use this hybrid framework. It suggests that incorporating known physics, even imperfect physics, helps create models that perform better in short-time predictions.

Lu: They found that PI-DManD-c and PI-DManD-B are robust to scarce data and FOM errors, proving the idea that blending data and physics is a powerful way to handle these kinds of systems.

Meng: From an engineering standpoint, even though training these hybrid methods takes more time than just using pure data methods, they still take a similar amount of time to evaluate them for testing, which means we get better short-time tracking for the same computational investment.

Lalam: The paper "Blending data and physics for reduced-order modeling of systems with spatiotemporal chaotic dynamics" shows that this hybrid method is robust to scarce data and FOM errors, demonstrating that including knowledge of the physics, even if inaccurate, helps in creating hybrid ROMs like PI-DManD-B that perform better than solely data-driven ROMs like DManD in short-time predictions.

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