Stable Long-Horizon Spatiotemporal Prediction on Meshes Using Latent Multiscale Recurrent Graph Neural Networks

1. Problem Statement

The paper addresses the challenge of accurate long-horizon spatiotemporal prediction of physical fields (specifically temperature) on complex, evolving geometries.

• Context: The primary application is Additive Manufacturing (AM), specifically metal powder bed fusion. In this process, the thermal history dictates defect formation, residual stresses, and mechanical properties.
• Challenges:
  • Computational Cost: High-fidelity numerical solvers (Finite Element Methods) are accurate but too slow for real-time inference or large-scale design exploration.
  • Long-Horizon Instability: Existing Machine Learning (ML) methods, particularly autoregressive models, suffer from error accumulation and instability when predicting over thousands of time steps.
  • Multiscale Dynamics: The physics involve coupled fast local phenomena (laser-material interaction, melt-pool formation) and slow global evolution (heat diffusion, interlayer cooling). Treating these in a single monolithic model often leads to training instability.
  • Irregular Meshes: Predictions must be made directly on variable, irregular simulation meshes rather than fixed grids, which complicates standard CNN/RNN approaches.

2. Methodology: Latent Multiscale Recurrent Graph Neural Network (LM-RGNN)

The authors propose a deep learning framework that decomposes the prediction task into two coupled models operating at complementary time scales, all operating directly on graph-structured mesh data.

A. Core Architecture Components

1. Graph-Structured Data: The model operates natively on the simulation mesh (nodes and edges), avoiding interpolation errors associated with fixed grids.
2. Variational Graph Autoencoder (VGAE):
   • Function: Encodes high-dimensional temperature fields defined on the mesh into a compact latent representation on a coarser, latent graph.
   • Benefit: Drastically reduces memory consumption and facilitates stable information propagation over long sequences. It uses hierarchical graph pooling/unpooling to capture multiscale geometric features.
3. Graph Gated Recurrent Unit (GraphGRU):
   • A recurrent unit adapted for graph data, replacing linear transformations with Message-Passing Graph Neural Networks (MPGNNs).
   • It models the temporal evolution of the latent state.
4. Truncated Backpropagation Through Time (TBPTT): Used to manage memory constraints during training on long sequences.

B. The Time-Multiscale Strategy

The framework explicitly separates the physics into two distinct temporal scales, trained as two independent but coupled models:

1. Interlayer Model (Slow Scale):
   • Goal: Predict the temperature field at the end of each printed layer.
   • Dynamics: Captures slow, diffusion-dominated thermal evolution and cumulative heat accumulation.
   • Input: Global geometric info and layer-level process descriptors (layer width, duration).
   • Output: The initial temperature state for the next layer.
2. Intralayer Model (Fast Scale):
   • Goal: Reconstruct the fine-grained temperature evolution within a single layer.
   • Dynamics: Captures fast, localized dynamics driven by laser motion and melt-pool phenomena.
   • Input: The initial temperature (predicted by the Interlayer model) and time-resolved laser parameters.
   • Output: The transient temperature sequence for that specific layer.

Inference Workflow: The Interlayer model predicts the state at layer boundaries. The Intralayer model then fills in the details between these boundaries. This allows for parallel computation of intralayer sequences once interlayer states are known, improving efficiency compared to strictly sequential full-length models.

3. Key Contributions

• Latent Multiscale Formulation: A novel strategy that decouples fast and slow dynamics into separate models, significantly improving training stability and long-horizon prediction accuracy compared to monolithic models.
• Direct Mesh Prediction: The framework operates directly on irregular, variable meshes using Graph Neural Networks, preserving geometric fidelity and avoiding interpolation artifacts.
• Memory Efficiency: By utilizing a VGAE to compress the state space, the model reduces memory footprint, enabling the training of long sequences (thousands of steps) on standard hardware.
• Generalizability: While tested on 2D AM simulations, the architecture is designed to be extensible to 3D geometries and other physics-driven systems with multiscale dynamics (e.g., fluid dynamics, climate modeling).

4. Experimental Results

The model was evaluated on a dataset of 140 simulated powder bed fusion scenarios (140 geometries, 6,600–12,000 time steps each).

• Performance Metrics:
  • Accuracy: The LM-RGNN achieved a Mean Absolute Error (MAE) of 6.50°C, significantly outperforming the baseline Decoupled-RGNN (26.57°C).
  • Gradient & Melt-Pool: It showed superior accuracy in predicting spatial/temporal gradients and melt-pool localization (mpIoU of 0.85 vs. 0.67 for baseline).
  • Temporal Stability: The model exhibited no noticeable error accumulation over 1,000+ time steps. In contrast, the baseline showed a persistent increase in error and temporal fluctuations.
• Efficiency:
  • Memory: The latent approach reduced memory pressure, allowing for larger batch sizes and stable training of long sequences.
  • Inference: The inference time was marginally higher (+0.97ms/step) than the baseline but provided vastly superior stability and accuracy.
• Ablation Studies:
  • Removing the latent space (operating on full fields) led to higher training variability and memory issues.
  • Removing the multiscale decomposition (training a single model on full sequences) resulted in poor gradient propagation and significantly degraded performance (MAE ~26°C).

5. Significance and Impact

• Scientific Machine Learning: This work advances the state-of-the-art in solving PDEs on graphs by demonstrating that latent multiscale modeling is a robust solution for long-horizon stability, a known bottleneck in autoregressive physics models.
• Additive Manufacturing: It offers a pathway to real-time process monitoring and control. By accurately predicting temperature histories and gradients, manufacturers can proactively adjust parameters to prevent defects (porosity, cracking) without relying on slow, expensive simulations.
• Broader Applicability: The methodology is applicable to any physical system characterized by coupled fast local and slow global dynamics on complex geometries, such as hemodynamic flow in arteries, turbomachinery flows, and climate/weather modeling.

In summary, the paper presents a robust, scalable, and stable framework for long-term spatiotemporal prediction on meshes, effectively solving the trade-off between computational efficiency, geometric fidelity, and prediction horizon length.