Learning Generalized Diffusions using an Energetic Variational Approach

TL;DR

This work addresses learning governing laws for dissipative systems by leveraging an energetic variational framework to identify the potential $\psi$ and the noise intensity $\sigma^2$ in generalized diffusions. By formulating a loss directly from the energy-dissipation law and representing $\psi$ with a neural network, the authors develop two data-compatibility approaches: a density-based method and a particle-to-density method. The methods demonstrate robustness to noisy data, scalability to higher dimensions, and the ability to learn from limited time-snapshot data, while preserving thermodynamic consistency via the fluctuation-dissipation relation. These contributions provide a principled, data-efficient pathway for thermodynamically consistent system identification with potential impact on fields where dissipative dynamics are essential.

Abstract

Extracting governing physical laws from computational or experimental data is crucial across various fields such as fluid dynamics and plasma physics. Many of those physical laws are dissipative due to fluid viscosity or plasma collisions. For such a dissipative physical system, we propose a framework to learn the corresponding laws of the systems based on their energy-dissipation laws, assuming either continuous data (probability density) or discrete data (particles) are available. Our methods offer several key advantages, including their robustness to corrupted/noisy observations, their easy extension to more complex physical systems, and the potential to address higher-dimensional systems. We validate our approaches through representative numerical examples and carefully investigate the impacts of data quantity and data property on model discovery.

Figures (8)

• Figure 1: The learned potential function $\psi_{nn}$ resulting from different number $M$ of groups of training data sets compared with the ground truth $\psi = 0.5 x^4 - x^2$ in the one-dimensional case of \ref{['eqn:generalized_diff']} with given noise intensity $\sigma(x)=\frac{1}{x^2+1}$. (a) Using the density-based method; (b) Using the particle-to-density method.
• Figure 2: (a) The learned potential function $\psi_{nn}$ resulting from training with different levels of noise and fixed number of groups of data ($M=15$) compared with the ground-truth potential $\psi$ and using the density-based method. (b) The relative $L_2$ difference $d_f(t)$\ref{['eqn:forward_error']} of the forward solutions to the Fokker-Planck equation \ref{['eqn:FPE_generalizedDiff']} using the learned potentials $\psi_{nn}$ and the ground truth $\psi$. The solutions of the Fokker-Planck equation \ref{['eqn:FPE_generalizedDiff']} using the learned potentials $\psi_{nn}$ with noise level $=0.0$ training data (c), noise level $=0.6$ (d) and the ground truth (e).
• Figure 3: (a) The evolution of the energy $E(t)$ for the case with $\psi= \frac{1}{2} x^4 - x^2 +x$ and $\sigma^2=(1+\cos(3x+\frac{1}{2}))^2$. (b) The learned potential functions $\psi$ using one group of training data ($M=1$) at an unsteady state ($t=20$) and at the steady state ($t=200$) compared with the ground truth potential $\psi$.
• Figure 4: The learned noise intensity $\sigma_{nn}^2$ resulting from different numbers of groups of training data sets $M$, compared with the ground truth $\sigma^2(x) = \frac{1}{(1+x^2)^2}$ in the one-dimensional case \ref{['eqn:generalized_diff']} with the given potential $\psi(x) = \frac{1}{2} x^4 - x^2$. (a) Using the density-based method; (b) Using the particle-to-density method.
• Figure 5: (a) Clean and corrupted training data. (b) The learned potential function $\psi_{nn}$ from the clean training data using our EnVarA-based method and the PDE-based method, compared with the true potential $\psi$. (c) The same as (b) except from the corrupted training data.

Theorems & Definitions (13)

• Remark 2.1
• Remark 2.2
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Example 4.1
• Example 4.2
• Example 4.3
• Example 4.4