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Learning two-phase microstructure evolution using neural operators and autoencoder architectures
Vivek Oommen, Khemraj Shukla, Somdatta Goswami, Remi Dingreville, George Em Karniadakis
TL;DR
Phase-field simulation is computationally expensive, and existing surrogate approaches face efficiency and reconstruction challenges. The paper combines a convolutional autoencoder with DeepONet to learn microstructure evolution in latent space and reconstruct future fields. The framework robustly supports solver replacement or acceleration, with a reported 29% speed-up in a hybrid setting.
Problem
Phase-field modeling captures mesoscale microstructure evolution but is computationally expensive, while RNN surrogates can be slow and require difficult reconstruction from statistical variables.
Method
A convolutional autoencoder compresses microstructures into a low-dimensional latent space, where DeepONet learns evolution mappings before decoding predicted fields.
Results
The framework robustly replaces the high-fidelity solver for interpolation or accelerates it for extrapolation, achieving a reported 29% speed-up in the hybrid approach.
Takeaways & Limitations
The autoencoder–DeepONet framework provides an efficient, accurate, and noise-robust route for predicting complex microstructural evolution.
Takeaways & Limitations
The approach does not initially resolve small-scale features during early fast dynamics, and the authors identify physics constraints and multi-fidelity data as future improvements.
Abstract
from arXiv · showhide
Phase-field modeling is an effective but computationally expensive method for capturing the mesoscale morphological and microstructure evolution in materials. Hence, fast and generalizable surrogate models are needed to alleviate the cost of computationally taxing processes such as in optimization and design of materials. The intrinsic discontinuous nature of the physical phenomena incurred by the presence of sharp phase boundaries makes the training of the surrogate model cumbersome. We develop a framework that integrates a convolutional autoencoder architecture with a deep neural operator (DeepONet) to learn the dynamic evolution of a two-phase mixture and accelerate time-to-solution in predicting the microstructure evolution. We utilize the convolutional autoencoder to provide a compact representation of the microstructure data in a low-dimensional latent space. DeepONet, which consists of two sub-networks, one for encoding the input function at a fixed number of sensors locations (branch net) and another for encoding the locations for the output functions (trunk net), learns the mesoscale dynamics of the microstructure evolution from the autoencoder latent space. The decoder part of the convolutional autoencoder then reconstructs the time-evolved microstructure from the DeepONet predictions. The trained DeepONet architecture can then be used to replace the high-fidelity phase-field numerical solver in interpolation tasks or to accelerate the numerical solver in extrapolation tasks.
INTRODUCTION
The paper addresses the computational difficulty of phase-field microstructure evolution by combining convolutional autoencoders with DeepONet to learn dynamics in a low-dimensional latent space. The resulting surrogate supports accurate forecasting, noise robustness, and hybrid acceleration of high-fidelity simulations.
- Motivation: Fourth-order Cahn–Hilliard dynamics combine rapid phase separation with slow coalescence, creating multiscale challenges for efficient and accurate numerical solution.The differing spatial and temporal scales make realistic-time simulation difficult.
- Motivation: RNN-based surrogates can predict microstructural statistics rapidly but remain slow for large datasets and require ill-posed microstructure reconstruction from non-primitive variables.The reconstruction can add errors when intricate interfacial length scales must be resolved.
- Framework: The proposed framework encodes microstructures into a compact latent space, applies DeepONet to learn their evolution, and decodes predictions back into the original microstructure space.DeepONet learns a function-to-function mapping from microstructure histories to future states.
- Model selection: Increasing latent dimension generally improves forecasting, with the best model using ld = 196, architecture 1, and a sine activation function.Models with tanh and sine activations performed better than those using ReLU for this problem.
- Accuracy: Forecasting error remains below 6%, while prediction error is higher during early rapid dynamics and improves as microstructural features coarsen.The model has difficulty resolving small-scale, high-frequency features at early times because of spectral bias.
- Robustness: The surrogate is nearly invariant to Gaussian white noise up to 10%, consistent with the autoencoder retaining dominant microstructural modes while filtering noise.The latent representation reduces the influence of unnecessary noise in the input data.
- Acceleration: The hybrid solver predicts ten time frames in 2 seconds after a 90-minute high-fidelity forecast, achieving a reported 29% speed-up and saving 135 minutes per evolution.The approach can replace the high-fidelity solver for interpolation or accelerate it for extrapolation.
DISCUSSION
The autoencoder–DeepONet framework learns microstructure dynamics in a compact latent space, offering efficient surrogate prediction while remaining data-driven and extensible. Its demonstrated scope includes interpolation, extrapolation, noisy data, higher-dimensional data, physics constraints, and multi-fidelity extensions.
- DISCUSSION: The framework combines convolutional autoencoders for nonlinear latent representations with DeepONet for learning microstructure dynamics in latent space.The model is trained on high-fidelity Cahn–Hilliard phase-field data.
- DISCUSSION: The trained model can replace the high-fidelity solver for interpolation or accelerate it for extrapolation, with increased latent dimension and emphasis on early time steps improving representation capability.These conclusions are reported for the demonstrated two-phase microstructure evolution framework.
- DISCUSSION: The latent-space approach avoids statistical-function post-processing and makes DeepONet training computationally efficient by smoothing high-gradient microstructure data.The encoder maps data from a difficult primitive-space distribution to a more gradual, trainable latent-space distribution.
- DISCUSSION: Although demonstrated on 2D data, the workflow can be extended using 3D convolutional and transpose-convolutional autoencoder layers.Each 3D realization would be represented as a sequence of 3D tensor data structures.
- DISCUSSION: The framework is purely data-driven and does not explicitly take boundary conditions as inputs; boundary-condition information is implicitly contained in the microstructure history supplied to DeepONet.The authors also note that DeepONet can map variable boundary conditions to output fields if desired.
- DISCUSSION: Future extensions include physics constraints and multi-fidelity fusion of phase-field simulations with experimental microstructure images.The proposed multi-fidelity direction treats numerical phase-field data as lower fidelity and experimental microscopy data as higher fidelity.
- DISCUSSION: Overall, the framework is presented as an efficient, rapid, and noise-robust alternative for predicting complex microstructural evolution relevant to materials discovery, understanding, optimization, and design.The authors characterize this as a computationally viable path for mesoscale evolutionary phenomena.
METHODS
The study models spinodal decomposition with a convolutional autoencoder and DeepONet, compressing 128×128 microstructures into a latent space where temporal dynamics are learned. The framework uses MEMPHIS-generated data and evaluates relative L2 prediction error across training, testing, and forecast times.
- Spinodal decomposition is modeled with a single order parameter representing solute concentration in a two-phase mixture.
- The dataset contains 2,000 microstructure evolution samples at 128×128 resolution, with 1,600 used for training and 400 for testing.Samples were generated by down-sampling 500 phase-field simulations.
- DeepONet encodes the 80-step latent history with a branch network and uses time as trunk input to predict the latent microstructure at desired times.The predicted latent state is decoded back into a 128×128 microstructure.
FIGURES
The figures present the autoencoder–DeepONet workflow, prediction quality, forecasting-error studies, feature recovery, latent-space smoothing, and hybrid integration with MEMPHIS. Together, they organize the model’s representation, accuracy, temporal sampling, and computational-acceleration analyses.
- Figure 1 shows autoencoder training followed by latent-space DeepONet training and decoder reconstruction of the predicted microstructure.The encoder processes the 80-step history, while DeepONet predicts the latent state at a requested time.
- Figure 2 compares true, predicted, and point-wise-error microstructures at t10, t30, and t99 using a model with ld = 196.
- Figure 3 plots relative L2 error D(t) over training-time steps and future time steps unseen during training.
- Figure 4 compares test forecasting error for training datasets sampled at four temporal spacings: 500k∆t, 2×500k∆t, 5×500k∆t, and 10×500k∆t.
- Figure 5 compares early-time microstructures predicted without and with repeated early-time samples to emphasize small features during training.
- Figure 6 contrasts high-gradient primitive microstructures with smoother latent representations and their corresponding density estimates.
- Figure 7 schematizes hybrid MEMPHIS–DeepONet prediction and reports computational time for each model together with error.
TABLES
The tables document the convolutional autoencoder hyperparameters, latent-dimension survey, and computational-time comparison between MEMPHIS and the hybrid model.
- Table 1 specifies the hyperparameters used in the convolutional autoencoder.
- Table 2 surveys latent dimension size, network architecture, and nonlinear activation functions.
- Table 3 compares computational time for MEMPHIS and the proposed hybrid model over one microstructure realization from t1 to t105.
Algorithms
Algorithm 1 alternates high-fidelity MEMPHIS simulation with DeepONet predictions, using the predicted state to leap forward through the evolution.
- The algorithm initializes the time-step counter and iterates until reaching the total number of time steps.
- MEMPHIS advances the initial state for nt time steps before DeepONet predicts the next DONnt time steps.
- The predicted state becomes the next MEMPHIS input, and the counter advances by the combined MEMPHIS-plus-DeepONet leap.
Supplementary Note 1: Principal Component Analysis (PCA) of microstructure dataset
The PCA analysis indicates that the microstructure dataset is non-diffusive, so retaining valuable information may require nonlinear rather than low-dimensional linear compression.
- Because PCA discards lower-eigenvalue modes, using few principal components could lose valuable information for this dataset.
- A nonlinear mapping from the high-dimensional data to a low-dimensional latent space is presented as an alternative compression strategy.
- Supplementary Figure 8 shows cumulative explained variance and energy curves for the dominant 1,000 microstructure eigenvalues.
- The two curves suggest that the microstructure data represent a non-diffusive system.
Supplementary Note 2: Experiment on number of training samples required
The sample-size experiment evaluates Autoencoder-DeepONet forecasting as training data increases from 25% to all 1,600 samples, with accuracy improving as sample count rises.
- Separate models were trained using 25%, 50%, 75%, and all 1,600 training samples.
- Increasing the number of training samples produced better accuracy in the model predictions.
- Model performance was evaluated using forecasting errors on test data at different time steps.
Supplementary Note 3: Experiment on choice of training loss function
The study compares five autoencoder training losses and then evaluates DeepONet forecasting using the resulting latent representations.
- Five autoencoder losses are compared: L1, relative L1, L2, relative L2, and a mixed loss.The mixed loss uses L2 for the initial 5,000 epochs and L1 thereafter.
- Forecasting error is evaluated through the relative L2 error Dtest(t) across test samples and time steps.
Supplementary Note 4: Error in forecasting
Forecasting performance is assessed by comparing mean relative L2 errors on training and test data across latent dimensions and future time frames.
- Mean relative L2 error is computed on both training and testing datasets for forecasting frames t90 through t99.
- Mean relative L2 error decreases as the autoencoder latent dimension increases.
Supplementary Note 5: Sensitivity to noise
The surrogate model is tested under increasing Gaussian input noise, with performance measured using relative L2 error across all time steps.
- Noise sensitivity is evaluated using Gaussian white noise with standard deviations from 0.5% to 10%.The tested levels are 0.5%, 1%, 2%, 3%, 4%, 5%, and 10%.
- The relative L2 norm does not increase much when noise is added to the model input.The authors attribute this behavior to autoencoder compression retaining dominant features and denoising the microstructure input.
Supplementary Note 6: Integrating DeepONet with MEMPHIS solver
A hybrid strategy combines autoencoder–DeepONet predictions with high-fidelity MEMPHIS simulations to accelerate phase-field predictions.
- The hybrid model uses autoencoder–DeepONet time leaps together with high-fidelity simulations.
- The comparison reports computational time for the autoencoder–DeepONet model and MEMPHIS, along with error on test data.