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Surrogate Models for Stochastic Al-Cu Phase-Field Solidification
Simon Savukoski, Tatu Pinomaa
TL;DR
Phase-field simulation is computationally costly, and stochastic surrogate prediction for coupled phase–solute evolution under moving thermal forcing has received limited attention. The paper evaluates VP-DDBM for state-conditioned Al–Cu prediction; it reports lower long-horizon errors on several quantities, retained dendritic structure, and faster inference than the solver.
Problem
Phase-field models make long trajectories and large parameter studies computationally expensive, while stochastic surrogate prediction for coupled phase–solute evolution under a moving temperature field has received limited attention.
Source: §1 · p. 1Method
Results
73× faster than the phase-field solver for equivalent 750-step advancement, VP-DDBM also yields lower errors across four quantities at the lower held-out gradient and three at the upper, while retaining densely branched morphology.
Source: §5 · p. 3Takeaways & Limitations
VP-DDBM demonstrates potential for faster process-condition screening and ensemble-based microstructure studies in stochastic Al–Cu solidification.
Source: §1 · p. 2, §5 · p. 3Evaluation is limited to held-out thermal-gradient families; broader thermal histories and processing conditions remain for future evaluation.
Source: §5 · p. 3
Abstract
from arXiv · showhide
Phase-field models resolve microstructural evolution during rapid alloy solidification, but their computational cost limits long trajectories, stochastic ensembles, and studies across process conditions. We present a variance-preserving denoising diffusion bridge model (VP-DDBM) for state-conditioned prediction of two-dimensional Al-Cu solidification. Given the current phase, concentration, and prescribed temperature fields, the model predicts the subsequent state and is applied autoregressively. Across two held-out thermal gradients, VP-DDBM produces lower long-horizon errors than deterministic U-Net and SI-ConvNeXt baselines in domain-mean phase, domain-mean Cu concentration, and leading-tip position. It also retains fine dendritic structure over long rollouts, whereas the deterministic U-Net progressively smooths the morphology. VP-DDBM predicts the phase and concentration evolution approximately $73\times$ faster than the phase-field solver, showing potential for faster process-condition screening and ensemble-based microstructure studies.
1 Introduction
Phase-field models capture alloy solidification but make long trajectories and large parameter studies computationally expensive. This work compares stochastic and deterministic surrogates for predicting Al–Cu solidification under moving thermal forcing.
- Resolving phase-field equations at required spatial and temporal scales makes long trajectories and large parameter studies computationally expensive.Source: §1 · p. 1
- Although machine-learning surrogates have been developed for phase-field evolution, stochastic prediction for coupled phase–solute evolution under a moving temperature field has received limited attention.Source: §1 · p. 1
- The study compares U-Net and SI-ConvNeXt with VP-DDBM for recursive next-state prediction from current phase and concentration fields and prescribed temperature fields.Source: §1 · p. 1
- Evaluation emphasizes physical morphology and bulk quantities over exact branch-by-branch agreement, with potential applications in process-condition screening and ensemble-based microstructure studies.Source: §1 · p. 2
2 Phase-field model for rapid solidification
The Al–Cu phase-field model couples phase and concentration evolution under a prescribed moving temperature field, following formulations for dilute-alloy solidification.
- The order-parameter equation describes phase evolution with anisotropic interface properties, a driving force, and Gaussian stochastic forcing [15].Source: §2 · p. 2
- The concentration field follows a nonlinear diffusion equation with liquid-phase solute diffusion, chemical potential, and an antitrapping-current prefactor for controlled rapid-solidification kinetics.Source: §2 · p. 2
- The prescribed temperature is a moving linear profile, and the simulator advances the equations with explicit Euler integration at constant Δt; implementation parameters are given in Ref..Source: §2 · p. 2
3 Surrogate models
The surrogates predict the next phase and concentration state from the current state and prescribed temperature. VP-DDBM learns a conditional distribution over possible next states using a variance-preserving stochastic bridge.
- All surrogates map the current phase and Cu-concentration fields plus prescribed temperature to the next stored state; temperature is not predicted.Source: §3 · p. 2
- The deterministic baselines are a residual encoder–decoder U-Net and SI-ConvNeXt, which uses resolution-preserving ConvNeXt blocks without an encoder–decoder bottleneck [20].Source: §3 · p. 2
- VP-DDBM trains on stochastic bridge states formed as noisy mixtures of known current and next states, conditioning next-state recovery on the current state, temperature, and bridge time.Source: §3 · p. 2
- At inference, DBIM traverses the learned bridge from the known current-state endpoint toward a sampled next state; with η = 0, remaining updates are deterministic.Source: §3 · pp. 2–3
- The U-Net denoisers combine global temperature conditioning through FiLM with multiresolution processing of the full temperature field, and VP-DDBM adds a bridge-time embedding.Source: §3 · p. 3
4 Training and evaluation
The evaluation uses trajectories spanning 14 thermal gradients, with complete gradient families held out for validation and testing. Models are assessed in autoregressive rollouts using physical quantities and timed against the solver.
- The dataset contains 70 independently generated trajectories on a 512 × 768 grid, spanning 14 thermal gradients and 26,870 adjacent transitions of 750 phase-field Euler steps each.Source: §4 · p. 3
- Complete gradient families are split before transition pairs are formed: ten gradients train the models, while two each are reserved for validation and testing.Source: §4 · p. 3
- At test time, predictions are recursively fed back while temperature advances; evaluation measures relative errors in domain-mean phase, domain-mean Cu concentration, interface length, and leading-tip position.Source: §4 · p. 3
- Runtime comparisons measure the same 750-step interval, timing the phase-field solver on 1,536 CPU ranks and surrogate inference on one AMD MI250X GCD.Source: §4 · p. 3
5 Results
Across long autoregressive rollouts, VP-DDBM has lower errors than the deterministic baselines on most reported quantities and retains densely branched morphology. It also advances phase and concentration faster than the phase-field solver, though broader thermal histories remain future work.
5 Results
- For 750 phase-field Euler steps, VP-DDBM takes 0.634 s versus 46.29 s for the solver, a reported wall-clock speedup of approximately 73×.The timing uses five DBIM network evaluations and the reported hardware configurations.Source: §5 · p. 3
- The authors identify broader thermal histories and processing conditions relevant to additive manufacturing as future evaluation settings.The conclusion characterizes acceleration of stochastic Al–Cu phase-field simulations as potential.Source: §5 · p. 3
- VP-DDBM yields lower errors than both deterministic baselines across all four quantities at the lower held-out gradient; at the upper gradient, this ordering holds for three quantities but not interface length.Figure 1 reports mean relative errors across 15 rollouts over 30,000 phase-field Euler steps at the lower gradient.Source: §5 · p. 3, Figure 1 · p. 4
- VP-DDBM reproduces similar front advancement and Cu redistribution while retaining a densely branched morphology in a 65-transition rollout.The rollout spans 48,750 phase-field Euler steps at the lower held-out gradient.Source: §5 · p. 3, Figure 2 · p. 4
A.1 Model and training hyperparameters
The selected production runs use a fixed AdamW training setup. VP-DDBM uses a variance-preserving bridge schedule and five-evaluation DBIM inference, with randomness introduced at initialization.
A.1 Model and training hyperparameters
- Selected production runs use AdamW, constant learning rates, an 800-epoch horizon, and an effective global batch size of 128.Source: §A.1 · p. 7
- VP-DDBM samples bridge times from U(10^-4, 1) with βmin = 0.1 and βd = 2.0, then uses DBIM with η = 0 and five network evaluations.The bridge-time grid is [0.999, 0.6660333, 0.3330667, 0.0001, 0].Source: §A.1 · p. 7
- A Gaussian perturbation initializes the first bridge time; subsequent DBIM updates are deterministic conditional on that perturbation.Source: §A.1 · p. 7
A.2 Evaluation details
Evaluation measures domain-mean phase and Cu concentration, interface length, and leading-tip position across 15 rollouts per held-out thermal gradient. Spatial rollouts and upper-gradient error curves provide complementary views of model behavior.
A.2 Evaluation details
- Domain-mean phase and Cu concentration are computed over grid points, while the two-dimensional interface is the phase field’s zero contour.Interface length is defined using one-dimensional Hausdorff measure; leading-tip position is measured along the growth direction.Source: §A.2 · p. 7
- Errors are averaged over 15 rollouts per held-out thermal gradient, using three starting states from each of five independently generated trajectories.The starting states are at simulator steps 220,000, 230,000, and 240,000.Source: §A.2 · p. 7
- Figure S1 compares phase-field reference and surrogate rollouts across a common initialization and three later states, showing phase and Cu concentration in separate blocks.The rollout ends after 65 autoregressive transitions; the VP-DDBM row shows one sampled realization.Source: Figure S1 · p. 8
- Figure S2 reports upper-gradient relative-error curves for four physical quantities over 30,000 phase-field Euler steps, with SI-ConvNeXt values clipped outside the displayed ranges.Curves average 15 held-out rollouts at G = 2.2×10^6 K m−1.Source: Figure S2 · p. 9