Training Neural Operators to Preserve Invariant Measures of Chaotic Attractors

Core Concepts

Training neural operators to preserve the invariant measures and time-invariant statistics of chaotic attractors, rather than focusing solely on short-term forecasting accuracy, enables more stable and physically relevant long-term predictions.

Abstract

The paper addresses the challenge of training neural operators to accurately emulate complex chaotic dynamical systems, where small perturbations in initial conditions lead to exponential divergence of trajectories over long time horizons.
The authors propose two novel approaches to preserve the invariant measures and time-invariant statistics of chaotic attractors during neural operator training:

Optimal Transport (OT) Loss: This approach uses expert knowledge to define a set of summary statistics that characterize the important physical properties of the system. The neural operator is then trained to match the distribution of these summary statistics between the model predictions and the observed data using an optimal transport loss.

Contrastive Learning (CL) Loss: In the absence of prior knowledge about relevant statistics, this approach uses self-supervised contrastive learning to automatically learn a set of time-invariant features that can distinguish between different chaotic attractors. The neural operator is then trained to preserve these learned features.

Both approaches are combined with a standard root mean squared error (RMSE) loss evaluated over a short time horizon to ensure the neural operator captures any remaining short-term predictability in the dynamics.
The authors demonstrate the effectiveness of their approaches on two chaotic dynamical systems - the Lorenz-96 model and the Kuramoto-Sivashinsky equation. They show that neural operators trained with the OT or CL losses significantly outperform a baseline trained only with RMSE in terms of preserving the invariant statistics, energy spectra, Lyapunov exponents, and fractal dimensions of the chaotic attractors, even in the presence of measurement noise.

Stats

The dynamics of the Lorenz-96 system are governed by the equation: dui/dt = (ui+1 - ui-2)ui-1 - ui + F, where F is a parameter that controls the chaotic behavior.
The dynamics of the Kuramoto-Sivashinsky system are governed by the equation: ∂u/∂t = -u∂u/∂x - φ∂²u/∂x² - ∂⁴u/∂x⁴, where φ is a parameter that controls the chaotic behavior.
Noisy measurements of the state variables u are used to train the neural operators, with the noise scale r varying from 0.1 to 0.3.

Quotes

"Chaos not only presents a barrier to accurate forecasts but also makes it challenging to train emulators, such as neural operators, using the traditional approach of rolling-out multiple time steps and fitting the root mean squared error (RMSE) of the prediction, as demonstrated in Fig. 1."
"By training a neural operator to preserve this invariant measure—or equivalently, preserve time-invariant statistics—we can ensure that the neural operator is properly emulating the chaotic dynamics even though it is not able to perform accurate long-term forecasts."

Key Insights Distilled From

Training neural operators to preserve invariant measures of chaotic attractors

by Ruoxi Jiang,... at arxiv.org 04-18-2024

https://arxiv.org/pdf/2306.01187.pdf

Deeper Inquiries

How could the proposed approaches be extended to handle explicit time dependence in the dynamical systems, such as time-dependent forcing or control parameters

To extend the proposed approaches to handle explicit time dependence in dynamical systems, such as time-dependent forcing or control parameters, we can modify the training process to incorporate this additional complexity. One way to achieve this is by considering a time window over which we compute the relevant statistics and select positive pairs for contrastive learning. By restricting the time range, we can capture the dynamics within that window and focus on the specific time-dependent features or parameters of interest. This approach allows us to study slowly varying dynamics or sharp discrete transitions, such as tipping points, by aligning the positive pairs within the defined time window. Additionally, for systems with explicit time dependence, we can introduce additional features or embeddings that encode the time-dependent information, enabling the neural operator to learn and preserve these temporal dynamics during training.

How sensitive are the contrastive learning results to the diversity of the environments present in the multi-environment training data, and how could the data augmentation be improved to increase this diversity

The sensitivity of contrastive learning results to the diversity of environments in the multi-environment training data is crucial for the effectiveness of the approach. A diverse set of environments ensures that the neural network learns robust and generalizable features that can distinguish between different trajectories and capture the underlying dynamics effectively. To improve the diversity of environments in the training data, we can employ various data augmentation techniques. For example, we can introduce random perturbations or transformations to the trajectories, simulate different environmental conditions, or vary the initial conditions and parameters across the training instances. By augmenting the data with a wide range of environmental settings, we can enhance the network's ability to learn invariant statistics that are representative of the entire system's dynamics, leading to more robust and accurate emulators.

Can the ideas of preserving invariant measures be combined with other recent advances in training neural operators, such as using Sobolev norms or dissipative regularization, to further improve the long-term stability and physical relevance of the emulators

The ideas of preserving invariant measures can indeed be combined with other recent advances in training neural operators, such as using Sobolev norms or dissipative regularization, to further improve the long-term stability and physical relevance of the emulators. By incorporating Sobolev norms or dissipative regularization into the loss function alongside the preservation of invariant measures, we can encourage the neural operator to capture not only the statistical properties of the dynamics but also the underlying physical constraints and regularities of the system. This combined approach can help the emulator better model the long-term behavior of chaotic systems, ensuring that the predictions are not only statistically accurate but also physically meaningful and consistent with the known dynamics of the system. By integrating multiple regularization techniques and constraints, we can create more robust and reliable neural operators for a wide range of applications in chaotic systems.