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Disentangled Latent Dynamics Manifold Fusion for Solving Parameterized PDEs

arXiv:2603.12676v1 Announce Type: new Abstract: Generalizing neural surrogate models across different PDE parameters remains difficult because changes in PDE coefficients often make learning harder and optimization less stable. The problem becomes even more severe when the model must also predict beyond the training time range. Existing methods usually cannot handle parameter generalization and temporal extrapolation at the same time. Standard parameterized models treat time as just another input and therefore fail to capture intrinsic dynamics, while recent continuous-time latent methods often rely on expensive test-time auto-decoding for each instance, which is inefficient and can disrupt continuity across the parameterized solution space. To address this, we propose Disentangled Latent Dynamics Manifold Fusion (DLDMF), a physics-informed framework that explicitly separates space, time, and parameters. Instead of unstable auto-decoding, DLDMF maps PDE parameters directly to a contin

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Zhangyong Liang, Ji Zhang · · 1 min read · 4 views
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arXiv:2603.12676v1 Announce Type: new Abstract: Generalizing neural surrogate models across different PDE parameters remains difficult because changes in PDE coefficients often make learning harder and optimization less stable. The problem becomes even more severe when the model must also predict beyond the training time range. Existing methods usually cannot handle parameter generalization and temporal extrapolation at the same time. Standard parameterized models treat time as just another input and therefore fail to capture intrinsic dynamics, while recent continuous-time latent methods often rely on expensive test-time auto-decoding for each instance, which is inefficient and can disrupt continuity across the parameterized solution space. To address this, we propose Disentangled Latent Dynamics Manifold Fusion (DLDMF), a physics-informed framework that explicitly separates space, time, and parameters. Instead of unstable auto-decoding, DLDMF maps PDE parameters directly to a continuous latent embedding through a feed-forward network. This embedding initializes and conditions a latent state whose evolution is governed by a parameter-conditioned Neural ODE. We further introduce a dynamic manifold fusion mechanism that uses a shared decoder to combine spatial coordinates, parameter embeddings, and time-evolving latent states to reconstruct the corresponding spatiotemporal solution. By modeling prediction as latent dynamic evolution rather than static coordinate fitting, DLDMF reduces interference between parameter variation and temporal evolution while preserving a smooth and coherent solution manifold. As a result, it performs well on unseen parameter settings and in long-term temporal extrapolation. Experiments on several benchmark problems show that DLDMF consistently outperforms state-of-the-art baselines in accuracy, parameter generalization, and extrapolation robustness.

Executive Summary

The article introduces Disentangled Latent Dynamics Manifold Fusion (DLDMF), a novel physics-informed framework for solving parameterized partial differential equations (PDEs). DLDMF tackles the challenge of generalizing neural surrogate models across different PDE parameters by explicitly separating space, time, and parameters. This approach improves performance on unseen parameter settings and long-term temporal extrapolation. Experiments demonstrate DLDMF's superiority over state-of-the-art baselines in terms of accuracy, parameter generalization, and extrapolation robustness. While the proposed method shows promise, its applicability to complex, real-world problems remains uncertain. Further investigation into its scalability and adaptability is necessary to fully realize its potential.

Key Points

  • ▸ DLDMF explicitly separates space, time, and parameters to improve performance on unseen parameter settings.
  • ▸ The method uses a feed-forward network to map PDE parameters directly to a continuous latent embedding.
  • ▸ DLDMF employs a dynamic manifold fusion mechanism to combine spatial coordinates, parameter embeddings, and time-evolving latent states.
  • ▸ The approach outperforms state-of-the-art baselines in accuracy, parameter generalization, and extrapolation robustness.

Merits

Improved Generalizability

DLDMF's separation of space, time, and parameters enables better generalization across different PDE parameters.

Enhanced Extrapolation Robustness

The method's ability to model prediction as latent dynamic evolution improves long-term temporal extrapolation robustness.

Demerits

Complexity

DLDMF's architecture, while innovative, may be challenging to implement and interpret, particularly for complex, real-world problems.

Scalability

The method's performance on large-scale problems remains uncertain, and further investigation into its scalability is necessary.

Expert Commentary

While DLDMF demonstrates impressive performance on benchmark problems, its applicability to real-world problems remains uncertain. The method's reliance on a feed-forward network to map PDE parameters to a continuous latent embedding may not be scalable to complex, high-dimensional problems. Furthermore, the dynamic manifold fusion mechanism, although innovative, may require significant computational resources to implement. Nevertheless, DLDMF's potential to improve generalizability and extrapolation robustness makes it an exciting development in the field of computational physics. Future research should focus on investigating the method's scalability and adaptability to complex, real-world problems.

Recommendations

  • ✓ Investigate the scalability of DLDMF on large-scale problems.
  • ✓ Further research is necessary to fully realize the method's potential and its applicability to complex, real-world problems.

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