Thermodynamics-informed super-resolution of scarce temporal dynamics data

TL;DR

This work tackles the challenge of predicting high-resolution temporal dynamics from scarce data by jointly learning a nonlinear reduced-order representation and a physically consistent time evolution. An adversarial autoencoder (AAE) performs model reduction and super-resolution, while a structure-preserving neural network (SPNN) evolves the latent variables under the GENERIC formalism to guarantee thermodynamic consistency (energy conservation and nonnegative entropy production). The approach is demonstrated on two cylinder-flow CFD cases, a Newtonian and a Herschel-Bulkley non-Newtonian fluid, showing accurate high-resolution reconstructions and rollout dynamics that outperform both bicubic interpolation and a black-box baseline. The results suggest a promising route for real-time digital twins where data are scarce, providing thermodynamically sound predictions and significant speedups over high-fidelity simulations.

Abstract

We present a method to increase the resolution of measurements of a physical system and subsequently predict its time evolution using thermodynamics-aware neural networks. Our method uses adversarial autoencoders, which reduce the dimensionality of the full order model to a set of latent variables that are enforced to match a prior, for example a normal distribution. Adversarial autoencoders are seen as generative models, and they can be trained to generate high-resolution samples from low-resoution inputs, meaning they can address the so-called super-resolution problem. Then, a second neural network is trained to learn the physical structure of the latent variables and predict their temporal evolution. This neural network is known as an structure-preserving neural network. It learns the metriplectic-structure of the system and applies a physical bias to ensure that the first and second principles of thermodynamics are fulfilled. The integrated trajectories are decoded to their original dimensionality, as well as to the higher dimensionality space produced by the adversarial autoencoder and they are compared to the ground truth solution. The method is tested with two examples of flow over a cylinder, where the fluid properties are varied between both examples.

Figures (15)

• Figure 1: Scheme of the proposed framework. First, an encoder is used to reduce the dimensionality of the problem, obtaining a set of reduced variables or latent code. Then, a structure-preserving neural network (SPNN) is trained to integrate the time evolution of the reduced variables of the system. Thus, given the state of the system at time instant $n$, the net obtains the state at time instant $n + \Delta n$. Finally the decoder is used to recover the data to its original dimensionality and to generate the output in a resolution that is higher that the input one.
• Figure 2: Scheme of the adversarial autoencoder (AAE). The autoencoder takes a snapshot of the simulation as input and learns an encoded representation of the data. The decoder recovers the data to its original dimensionality, $\hbox{\boldmath$\hat{x}$}_{\text{AAE}}$, and it is trained to generate a higher-resolution output, $\hbox{\boldmath$\hat{X}$}_{\text{AAE}}$, achieving "superresolution" of the analysed simulation. The discriminator takes as input a sample that follows the proposed distribution or prior (in this work a normal distribution) and the latent code generated by the autoencoder. The discriminator output is in the range between 0 and 1. An output close to 0 means that the latent code does not match the prior distribution, while if the output is close to 1 the autoencoder is able to produce a latent code that fits into the prior distribution.
• Figure 3: Scheme of the structure-preserving neural network (SPNN). The SPNN is trained to predict the full time evolution of the latent variables generated by the AAE by applying the GENERIC structure of the underlying physics of the problem. The network takes the current snapshot as input and outputs the L and M matrices, as well as the energy and entropy gradients. Then, they are integrated following the GENERIC formalism, as shown in Eq. \ref{['eq:generic_integration']}, and the latent variables of the next snapshot are obtained. This process can be done iteratively, obtaining the rollout prediction of the full simulation.
• Figure 4: Training and validation loss curves for the Adversarial Autoencoder (Left) and the SPNN (right).
• Figure 5: Results of the prediction made by the Adversarial Autoencoder (AAE). \ref{['fig:aae_example_1_newt_lr']}: Low resolution Ground Truth (GT), AAE prediction and absolute error for $P$, $U_{x}$ and $U_{y}$. The ground truth is the input of the AAE and it predicts the low resolution fields and the high resolution fields shown in Fig. \ref{['fig:aae_example_1_newt_hr']}. \ref{['fig:aae_example_1_newt_hr']}: High resolution GT, AAE prediction and absolute error fields for the same snapshot shown in Fig. \ref{['fig:aae_example_1_newt_lr']}. \ref{['fig:aae_example_2_newt_lr']}: Low resolution fields for a different snapshots. \ref{['fig:aae_example_2_newt_hr']}: High resolution GT, AAE pred. and abs. error fields for the snapshot shown in Fig. \ref{['fig:aae_example_2_newt_lr']}.