Deep Onsager–Machlup Method (DOMM)

• Deep Onsager–Machlup Method (DOMM) is a mesh-free, DNN-parameterized variational approach to simulate irreversible soft-matter dynamics using the Onsager–Machlup principle.
• DOMM leverages residual neural networks and adaptive time-splitting to efficiently solve nonlinear, multi-scale, and multiphysics problems with an explicit thermodynamic structure.
• Demonstrations on diffusion, Cahn–Hilliard, and coupled Stokes dynamics validate DOMM’s accuracy and stability, achieving error thresholds and reducing training steps by 20–30%.

The Deep Onsager–Machlup Method (DOMM) is a computational framework for modeling soft matter dynamics. It synthesizes deep neural networks (DNNs) with the Onsager–Machlup variational principle (OMVP), enabling efficient, physics-informed solutions to nonlinear, multi-scale, and multiphysics problems characterized by multiple slow variables and dissipative processes. By replacing traditional hand-crafted ansatz functions with DNN-parameterized trial solutions, DOMM provides a mesh-free, variationally consistent strategy for simulating irreversible dynamics with explicit thermodynamic structure and high-order differential operators. DOMM can be viewed as an extension of the deep Ritz method, traditionally applied to elliptic and static problems, to dynamic settings governed by OMVP (Li et al., 2023).

1. Theoretical Foundations

DOMM is founded on the Onsager–Machlup variational principle, which provides a path-probability functional for the evolution of coarse-grained (slow) state variables $\alpha \equiv (\alpha_1, \alpha_2, \ldots)$. For dynamics over $[t_0, T]$ with rates $\dot{\alpha}$ and conjugate generalized forces $f(\alpha, t)$, the Onsager–Machlup action is

$$P[\alpha(t)] \propto \exp\left\{-\frac{1}{2k_BT} \mathcal{L}[\alpha(t)]\right\}$$

where the OM functional

$$\mathcal{L}[\alpha] = \int_{t_0}^T dt\, \frac{1}{2} (\dot{\alpha} - \dot{\alpha}^*)^T \zeta(\alpha) (\dot{\alpha} - \dot{\alpha}^*)$$

with $\dot{\alpha}^* \equiv \mu(\alpha) f(\alpha, t)$ and $\zeta = \mu^{-1}$. In systems where the free energy $F(\alpha)$ defines conservative forces $f_c=-\nabla_\alpha F$ and additional non-conservative (active) forces $f_a$ may be present, this becomes

$$\mathcal{L}[\alpha] = \int_{t_0}^T dt\, \left[ \frac{1}{2} \dot{\alpha}^T M^{-1}(\alpha) \dot{\alpha} + \frac{1}{2} \nabla F(\alpha)^T M(\alpha) \nabla F(\alpha) \right]$$

where $M = \zeta^{-1}$ is the mobility. Extremizing $\mathcal{L}$ via the Euler–Lagrange equation yields the irreversible force-balance or gradient flow equations, e.g.,

$$\dot{\alpha} = -M(\alpha) \nabla F(\alpha) + \ldots$$

2. DOMM Formulation

The distinctive feature of DOMM is the use of DNNs to parameterize the unknown dynamic field(s), typically denoted $\varphi(x,t)$ for order parameters, fluxes, or auxiliary quantities. The trial solution is expressed as $\varphi_\theta(x, t)$ where $\theta \in \mathbb{R}^N$ encompasses all network weights and biases. The architecture is generally a residual neural network (ResNet), described by

$$s^{0} = (x, t); \quad s^{\ell+1} = s^{\ell} + \sigma(W_\ell s^\ell + b_\ell), \quad \ell=0, ..., N_d-1; \quad \varphi_\theta(x,t) = a \cdot s^{N_d} + b$$

with $\sigma(s) = \tanh^3(s)$ as a smooth activation function to enable exact computation of high-order derivatives.

The loss functional minimized during training is

$$\mathcal{L}[\theta] = w_\text{om} \mathcal{L}_\text{om}[\varphi_\theta] + w_\text{ic} \mathcal{L}_\text{ic}[\varphi_\theta] + w_\text{bc} \mathcal{L}_\text{bc}[\varphi_\theta] + w_\text{con} \mathcal{L}_\text{con}[\varphi_\theta]$$

• $\mathcal{L}_\text{om}$: Onsager–Machlup (OM) action integral expressing thermodynamic irreversibility.
• $\mathcal{L}_\text{ic}$: Quadratic penalty enforcing initial conditions.
• $\mathcal{L}_\text{bc}$: Quadratic penalty enforcing boundary conditions (Dirichlet, Neumann, or periodic).
• $\mathcal{L}_\text{con}$: Penalty enforcing auxiliary constraints (mass conservation, incompressibility, etc.).

For order-parameter-conserving problems, $\mathcal{L}_\text{om}$ is

$$\mathcal{L}_\text{om} = \int_0^T dt \int_\Omega dx\, \frac{1}{2} (J - J^*)^T M^{-1}(\varphi) (J - J^*)$$

where $J^*$ is the variationally optimal flux (e.g., $-M \nabla \mu$ in Cahn–Hilliard models).

3. Network Architecture, Collocation, and Training Procedure

The DOMM deploys fully connected ResNet architectures with depth $N_d \sim 3$–$6$ and width $N_w \sim 50$–$128$. The choice of cubic hyperbolic tangent activations $\sigma(s) = \tanh^3(s)$ ensures regularity of high-order derivatives, facilitating treatment of fourth-order operators.

Training is performed over a mesh-free set of collocation points $(x_i, t_i)$ sampled uniformly in $(\Omega, [0,T])$. To improve computational scalability and leverage the “frequency principle” in DNN optimization, the solution domain in time is split into intervals of length $\Delta t$, and training proceeds sequentially over $[0, \Delta t], [\Delta t, 2\Delta t], \ldots$ This splitting reduces GPU memory requirements and preferentially optimizes smoother portions of the solution trajectory.

Mini-batching is employed by randomly sampling $N_k$ points from the collocation set per stochastic gradient descent (SGD) step. The Adam optimizer is used with learning rate $\eta \sim 10^{-3}$–$10^{-4}$. Loss weights $w_{\cdot}$ are empirically tuned to ensure comparable magnitudes among terms; initial and constraint penalties typically require larger weights. Convergence is monitored by both training loss and mean-square-error (MSE) at reference validation points, with rigorous criteria: $\textrm{MSE}<10^{-5}$ for straightforward diffusion/Cahn–Hilliard problems and $\lesssim 10^{-3}$ for coupled system dynamics.

4. Illustrative Applications

Three representative classes of soft matter PDEs were studied to validate DOMM (Li et al., 2023):

4.1 1D Particle Diffusion

For $\partial_t n - D \,\partial_x^2 n = 0$, trials with both step and Fourier-mode initial data and Neumann or Dirichlet BCs achieved $\mathrm{MSE} < 10^{-5}$ in $\lesssim 2 \times 10^4$ training steps (ResNet: $N_w=50$, $N_d=6$, tanh$^3$ activation). Time-splitting with $\Delta t = 0.5\tau_0$ ($\tau_0 = L^2/D$) and reduced data-per-step led to a $20$–$30\%$ reduction in steps to reach the same accuracy.

4.2 Two-Phase Cahn–Hilliard Dynamics

1D Cahn–Hilliard: For

$$\partial_t \phi - M \partial_x^2[a\phi(\phi^2-1) - K \partial_x^2\phi]=0$$

with periodic boundaries and trigonometric initial data, DOMM achieved $\mathrm{MSE} < 10^{-5}$ with $N_w=128$, $N_d=3$, and $\Delta t = 0.1\tau$.

2D Droplet Coalescence: For

$$\partial_t \phi + \nabla \cdot J = 0; \quad J = -M \nabla \mu; \;\; \mu = a \phi (\phi^2 - 1) - K \nabla^2 \phi$$

on $[{-L}, L]^2$ with periodic BCs, initial data with two droplets, MSE$<10^{-5}$ was achieved using similar network settings; pointwise errors are $\lesssim 10^{-3}$ near interfaces.

4.3 2D Stokes–Cahn–Hilliard Dynamics (Coupled Hydrodynamics)

PDEs:

$$\partial_t \phi + \nabla\cdot(\phi v + J) = 0; \quad J = -M\nabla \mu$$

$$-\nabla p + \eta \nabla^2 v + \mu \nabla \phi = 0; \quad \nabla\cdot v=0$$

Using $N_w=128$, $N_d=5$, $\Delta t=0.1\tau$, and weighted constraints, solution phase-fields and velocities matched finite-difference methods (FDM; $\Delta x = \Delta y = 0.005L$) with local errors $\lesssim 1\%$ and MSE$_\phi,$MSE$_v<10^{-3}$ in $\sim 2 \times 10^4$ steps.

5. Advantages, Limitations, and Comparison

Key features and outcomes demonstrated for DOMM (Li et al., 2023) include:

• Mesh-free, continuous representation, eliminating the need for spatial meshes and naturally accommodating complex geometries and dynamic interfaces.
• OM-based variational loss allows consistent coupling of multiple physical processes with thermodynamically derived weights—no manual term-scaling is required.
• High-order spatial and temporal derivatives (e.g., up to fourth order as in Cahn–Hilliard) are handled via automatic differentiation, circumventing the need for explicit finite-difference stencils.
• Time-splitting leverages the DNN “frequency principle” to focus on lower-frequency (smoother) solution components first, reducing long-time training costs.
• In comparison to the deep Ritz method (static problems) and PINNs (physics-informed neural networks), DOMM uniquely incorporates energy dissipation structures and most-probable-path criteria from OMVP, resulting in enhanced stability and fidelity for irreversible dynamics.
• Major limitations include scaling of training cost with input dimensionality and network size, sensitivity to relative weighting of loss terms, and challenges for very stiff or nonlinear kinetics, which may require adaptive sampling or curriculum-learning strategies.

6. Extensions and Future Directions

DOMM's variational strategy admits several extensions:

• Inverse Problems: Calibration of free-energy landscapes and mobility coefficients through minimization of $\mathcal{L}[\theta]$ with respect to experimental or trajectory data.
• Rare-Event Pathways: Identification of minimum-action (most-probable-path) solutions for nucleation, macromolecular dynamics, and related rare-event problems.
• Active Matter: Extension of OMVP to systems with non-conservative active forces $f_a$, enabling learning of active transport coefficients and emergent flow characteristics.
• Multi-Scale Coupling: Embedding DOMM within hierarchical or domain-decomposition frameworks to address systems with disparate temporal or spatial scales, such as nematohydrodynamics or elasto-diffusion in gels.

DOMM provides a general, physics-informed, variationally consistent computational methodology for irreversible soft-matter dynamics, with rigorous enforcement of thermodynamic structure and verified predictive accuracy on canonical problems (Li et al., 2023).