Physics-guided Convolutional Neural Network for Domain Growth Prediction in Systems with Conserved Kinetics

Technical Summary: Physics-Guided Convolutional Neural Network for Domain Growth Prediction

Problem Statement
The spatiotemporal evolution of complex physical systems, such as phase separation in binary mixtures, is governed by nonlinear partial differential equations (PDEs), specifically the Cahn–Hilliard (CH) equation. Traditional numerical solvers (e.g., finite-difference, finite-element) are computationally expensive and face scalability challenges on modern architectures. While machine learning (ML) offers surrogate modeling alternatives, existing approaches face specific limitations when applied to conserved kinetics:

• Physics-Informed Neural Networks (PINNs): Struggle with the fourth-order spatial derivatives inherent in the CH equation, leading to unstable gradient propagation. They also suffer from spectral bias, favoring smooth solutions over the sharp interfaces typical of phase separation, and often fail to naturally enforce the conservation of the order parameter.
• Data-Driven Surrogates (e.g., RNNs, standard CNNs): While efficient, these models often fail to preserve the conservation of the order parameter during long-term autoregressive rollouts. Small prediction errors accumulate over time, leading to a gradual loss of the conserved quantity and physically inconsistent dynamics.

Methodology
The authors propose an attention-based, physics-guided convolutional neural network designed specifically to model systems with conserved kinetics. The methodology integrates deep learning architecture with physical constraints:

1. Architecture: The model is based on a Residual U-Net architecture, comprising a contracting path, a central bridge, and an expansive path.

   • Residual Blocks: Each path utilizes residual blocks containing group normalization (GN), GeLU activation, and convolutional layers with circular padding to facilitate deep network training.
   • Attention Mechanism: A self-attention block is embedded in the central bridge at the lowest spatial resolution (highest receptive field). This allows the network to capture global spatial correlations and large-scale morphological evolution, complementing local convolutional features.
   • Upsampling/Downsampling: The network employs strided convolutions for downsampling and transposed convolutions for upsampling, with skip connections to preserve spatial details.

2. Physics-Guided Loss Function: To enforce the intrinsic conservation law of the CH equation, the training objective includes a composite loss function:

   • Data Fidelity: Mean Squared Error (MSE) between the predicted and ground-truth fields.
   • Conservation Constraint: An additional penalty term ($L_{\langle\psi\rangle}$) that minimizes the squared deviation between the spatial mean of the predicted field and the ground truth. This explicitly ensures the global conservation of the order parameter ($\psi$) throughout the evolution.
   • Total Loss: $L = L_{MSE} + \alpha L_{\langle\psi\rangle}$, where $\alpha$ controls the regularization strength.

3. Training Protocol:

   • Data Generation: Training data was generated by numerically solving the dimensionless CH equation using a finite-difference method on a $64 \times 64$ lattice.
   • Dataset: The dataset includes 20 independent trajectories covering critical ($\psi_0 = 0$) and off-critical ($\psi_0 = \pm 0.2, \pm 0.4$) compositions over a time interval $t \in [0, 300]$.
   • Task: The model is trained to predict a snapshot at $t + 100\Delta t$ given a snapshot at $t$. Long-term evolution is generated via recursive autoregressive inference.

Key Results
The proposed framework was evaluated on both critical and off-critical mixtures, demonstrating high fidelity in capturing complex dynamics:

• Conservation of Order Parameter: The model successfully maintains the spatially averaged order parameter ($\langle\psi\rangle$) constant over time for both critical and off-critical compositions, effectively preserving the global conservation law even during long-time rollouts.
• Morphological Accuracy:
  • Critical Mixtures: The model accurately reproduces the bicontinuous morphology and domain coarsening. Errors are primarily confined to domain boundaries and accumulate very slowly, maintaining stability over long horizons.
  • Off-Critical Mixtures: While accuracy degrades slightly faster than in critical cases due to the complexity of droplet coalescence and breakup, the model still captures the overall droplet morphology and remains stable over extended time scales.
• Generalization: The model, trained on $64 \times 64$ systems, successfully predicts dynamics for larger system sizes ($160, 192, 256$) without noticeable degradation in accuracy.
• Physical Consistency:
  • Growth Laws: The predicted domain growth follows the Lifshitz–Slyozov law ($L(t) \sim t^{1/3}$) with high accuracy, matching numerical solvers for both critical and off-critical cases.
  • Dynamic Scaling: The model reproduces the dynamical scaling hypothesis, where correlation functions $C(r,t)$ collapse onto a universal master curve when plotted against the rescaled distance $r/L(t)$.
  • Statistical Distributions: The probability distribution of the order parameter $P(\psi)$ matches numerical results, correctly capturing the symmetry in critical mixtures and the asymmetry in off-critical mixtures.

Significance and Claims
The paper claims that the proposed framework offers a robust, data-driven alternative for modeling systems with conserved kinetics. Its primary significance lies in the explicit integration of physical conservation laws into the loss function, which prevents the accumulation of unphysical errors common in standard autoregressive ML models.

The authors emphasize that the model does not merely achieve pixel-to-pixel accuracy but successfully learns the underlying statistical and dynamical behaviors of the system. This includes:

1. Preserving the conserved quantity (order parameter) over long time scales.
2. Capturing the correct growth kinetics (Lifshitz–Slyozov law).
3. Reproducing dynamical scaling and morphological evolution.

The work suggests that this approach can be extended to other complex dynamical systems governed by conserved kinetics, provided the specific conservation law is known and can be encoded into the training objective. The authors note that future work could aim to learn these conservation laws directly during training, rather than requiring them as prior knowledge.