Deep learning-based computational method for soft matter dynamics: Deep Onsager-Machlup method

TL;DR

The paper presents the deep Onsager-Machlup method ($DOMM$), a unified framework that marries the Onsager-Machlup variational principle ($OMVP$) with deep neural networks to solve soft matter dynamics. By expressing the state as a neural-network trial function and minimizing a composite OM-based loss, $DOMM$ handles multi-physics and multi-scale problems without meshing, demonstrated on diffusion and two-phase dynamics (with and without hydrodynamics). The results show high accuracy against analytic or finite-difference solutions and favorable efficiency relative to PINN, including successful extensions to high dimensions and coupled hydrodynamic problems. The work underscores the potential of variational-DNN approaches for complex non-equilibrium soft matter systems and outlines future directions in inverse problems, rare-event analysis, and broader dynamics.

Abstract

A deep learning-based computational method is proposed for soft matter dynamics -- the deep Onsager-Machlup method (DOMM). It combines the brute forces of deep neural networks (DNNs) with the fundamental physics principle -- Onsager-Machlup variational principle (OMVP). In the DOMM, the trial solution to the dynamics is constructed by DNNs that allow us to explore a rich and complex set of admissible functions. It outperforms the Ritz-type variational method where one has to impose carefully-chosen trial functions. This capability endows the DOMM with the potential to solve rather complex problems in soft matter dynamics that involve multiple physics with multiple slow variables, multiple scales, and multiple dissipative processes. Actually, the DOMM can be regarded as an extension of the deep Ritz method (DRM) developed by E and Yu that uses DNNs to solve static problems in physics. In this work, as the first step, we focus on the validation of the DOMM as a useful computational method by using it to solve several typical soft matter dynamic problems: particle diffusion in dilute solutions, and two-phase dynamics with and without hydrodynamics. The predicted results agree very well with the analytical solution or numerical solution from traditional computational methods. These results show the accuracy and convergence of DOMM and justify it as an alternative computational method for solving soft matter dynamics.

Figures (7)

• Figure 1: Schematic illustration of the deep Onsager-Machlup method based on deep neural networks. The structure of the deep neural network with a depth $N_{\mathrm{d}}$ (i.e., consisting of $N_{\mathrm{d}}$ hidden layers) and width $N_\mathrm{w}$ is shown, where the hidden layers are connected by one linear transformation, activation, and a residual connection as explained in Eq. (\ref{['Eq:DOMM-MLNet']}). The input of the neural network is the temporal node $t$ and the three-dimension (3D) spatial coordinates $\mathbf{x}=\left\{x,y,z\right\}$. The output is the solution of a $n$-dimension set of slow variables $\boldsymbol{\alpha}(\mathbf{x},t)=\{\alpha_1, \ldots, \alpha_n\}$. In the last operation layer, the corresponding (spatial and/or temporal) derivatives are computed through the chain rule of automatic differentiation by autograd in pytorch.
• Figure 2: Application 1 -- Solving 1D diffusion dynamics: comparison between the trained solution using DOMM (open circles, no time-splitting) and the exact solutions (solid lines) in Eqs. (\ref{['Eq:App1-Sol1']}, \ref{['Eq:App1-Sol2']}) upon different initial and boundary conditions. (a1) Trained solution with an initial condition of a step function ${n}\left({x,0}\right)=0.5+0.45 \, \mathrm{sgn}\left({x}\right)$ and Neumann boundary condition at $x=\pm L$. (b1) Trained solution with an initial condition of sums of sinusoidal functions ${n}\left({x,0}\right)=\Sigma_{k=1}^{4} \sin(k\pi {x})$ and Dirichlet boundary condition at $x=\pm L$. The time $t$ is normalized by the unit $\tau_0=L^2/D$ and $\tilde{t}=t/\tau_0$. (a2) and (b2) show the convergence of the training loss $\mathcal{L}$ as a function of training steps for the solutions in (a1) and (b1), respectively. (a3) and (b3) display the convergence of Mean Squared Error (MSE) of particle density $n$ at different times as a function of training steps for the solutions to the two initial-boundary value problems. All the trained solutions are obtained using the same neural network structure with width $N_{\mathrm{w}}=50$, depth $N_{\mathrm{d}} = 6$.
• Figure 3: Solving high-dimensional diffusion dynamics: comparison between the trained solution using DOMM (open circles, no time-splitting) and the analytical solutions (solid lines) in Eq. (\ref{['Eq:App1.5-Sol']}) upon the initial condition in Eq. (\ref{['Eq:App1.5-n0']}) and the Dirichlet boundary condition of $n(\{x_i=\pm L\},t)=0$ ($i=1,..,d$) for $\tilde{t}=t/\tau_0\in [0.0,0.02]$. (a) Slice plots of the trained solution and the analytical solution in the 5-dimensional system (by setting the value of $x_i$ non-zero only in one of the 5 dimensions). (b) The MSE convergence (or the accuracy) of the trained solutions at 1- to 5-dimensional systems is plotted as a function of the number of training steps. All the trained solutions are obtained using the same DNN structures and parameters (depth $5$, width $200$, learning rate $0.016$, dataset $10^4$).
• Figure 4: Comparisons of different sampling strategies used in the DNN methods for solving PDEs. (a) Discrete sampling on the spatiotemporal meshes as in the traditional finite difference method (FDM) and finite element method (FEM) ChunLiu2022Reina2021VONN. (b) Random (uniform) sampling in spatial coordinates on a discrete mesh of small time steps: mesh-free only in space. (c) Random (uniform) sampling in both spatial and temporal coordinates: mesh-free in both space and time. (d) Random (uniform) sampling in both spatial and temporal coordinates. Multi-step training is carried out by splitting the whole time period into several time intervals.
• Figure 5: Application 2 -- Solving 1D two-phase dynamics. (a) Comparison between the trained solution using DOMM (dashed lines) and the numerical solution using the traditional FDM (solid lines) upon the initial condition $\phi(x;t=0)=-\cos (2 \pi x/L)$ and periodic boundary condition at $x=\pm L$. The time $t$ is normalized by the unit $0.01\tau$ with $\tau\equiv L^2/Ma$ and the interface thickness is taken to be $\epsilon = 10^{-2}L$. (b) Convergence of the training error (Mean Square Error, MSE) of $\phi$ (at the corresponding time in (a)) as a function of the training steps. (c) Trained solution of $\phi$ as a function of position $\tilde{x}=x/L$ and time $\tilde{t}=t/\tau$. (d) The pointwise error maps of the trained solution of $\phi$ is calculated using $\phi_{\mathrm{DOMM}}(x,t) - \phi_{\mathrm{FDM}}(x,t)$ as a function of position and time with of $\phi_{\mathrm{DOMM}}$ being the trained solution by DOMM and $\phi _{\mathrm{FDM}}$ being the numerical solution by the traditional FDM. All the trained results are obtained using the same neural network structure with width $N_{\mathrm{w}}=128$, depth $N_{\mathrm{d}}=3$.