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SineNet: Learning Temporal Dynamics in Time-Dependent Partial Differential Equations (2403.19507v1)

Published 28 Mar 2024 in cs.LG

Abstract: We consider using deep neural networks to solve time-dependent partial differential equations (PDEs), where multi-scale processing is crucial for modeling complex, time-evolving dynamics. While the U-Net architecture with skip connections is commonly used by prior studies to enable multi-scale processing, our analysis shows that the need for features to evolve across layers results in temporally misaligned features in skip connections, which limits the model's performance. To address this limitation, we propose SineNet, consisting of multiple sequentially connected U-shaped network blocks, referred to as waves. In SineNet, high-resolution features are evolved progressively through multiple stages, thereby reducing the amount of misalignment within each stage. We furthermore analyze the role of skip connections in enabling both parallel and sequential processing of multi-scale information. Our method is rigorously tested on multiple PDE datasets, including the Navier-Stokes equations and shallow water equations, showcasing the advantages of our proposed approach over conventional U-Nets with a comparable parameter budget. We further demonstrate that increasing the number of waves in SineNet while maintaining the same number of parameters leads to a monotonically improved performance. The results highlight the effectiveness of SineNet and the potential of our approach in advancing the state-of-the-art in neural PDE solver design. Our code is available as part of AIRS (https://github.com/divelab/AIRS).

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Citations (3)
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Summary

  • The paper introduces SineNet, a multi-stage architecture that mitigates temporal misalignment in U-Net skip connections for solving time-dependent PDEs.
  • It demonstrates superior one-step prediction accuracy and long-term stability across challenging benchmarks like Navier-Stokes and shallow water equations.
  • Ablation studies reveal that wave residuals and multi-wave processing are key to capturing dynamic temporal evolution in neural PDE solvers.

SineNet: A Multi-Scale and Multi-Stage Approach for Time-Dependent PDE Solving

Introduction

The development of neural network-based solvers for time-dependent partial differential equations (PDEs) has emerged as a significant research interest, motivated by applications across various scientific and engineering domains. Existing studies predominantly utilize the U-Net architecture, given its proficiency in multi-scale processing. However, these approaches face limitations due to temporally misaligned features within skip connections, a consequence of feature evolution across layers. Addressing this, we introduce SineNet, designed to mitigate misalignment through its novel architecture consisting of multiple U-Net blocks or "waves". This blog post provides an overview of our proposed method and its evaluation on several PDE datasets.

Model Architecture

U-Net Limitations and the Advent of SineNet

The inherent feature misalignment in U-Net skip connections arises from the discrepancy in temporal alignment between the input and target features. To counteract this, SineNet employs a multi-stage architecture where each wave handles a fraction of the latent temporal evolution, thereby reducing misalignment within stage-wise skip connections and enhancing overall performance.

SineNet's Structure

SineNet is articulated through the sequential connection of U-shaped blocks, with each tasked to progress the latent representation of the system's dynamic state incrementally. This enables a controlled evolution of features, aligning temporally closer to their corresponding targets at each stage. Moreover, SineNet leverages both parallel and sequential multi-scale processing mechanisms, facilitated by block residuals and architectural modifications to reduce feature map misalignment issues further.

Handling Boundary Conditions

Crucial to the performance of any neural PDE solver is the encoding of boundary conditions. SineNet applies different padding strategies depending on the nature of the boundary condition for the problem at hand, demonstrating flexibility in incorporating physical constraints of the system being modeled.

Empirical Evaluation

Benchmark Datasets

SineNet's effectiveness was benchmarked across three PDE datasets: incompressible Navier-Stokes equations, compressible Navier-Stokes equations, and shallow water equations. Each dataset presented its unique challenges, ranging from multi-scale phenomena to complex advection-diffusion interactions, serving as a robust testbed for our method.

Comparison and Results

SineNet demonstrated superior performance in both 1-step prediction accuracy and long-term rollout stability across all evaluated datasets, outperforming existing state-of-the-art methods. Notably, increasing the number of waves in SineNet, while maintaining the same parameter budget, introduced consistent gains in modeling accuracy, underscoring the effectiveness of the proposed architecture in handling temporal dynamics and feature misalignment.

Insights from Ablation Studies

Our ablation studies confirmed the critical role of wave residuals in mitigating misalignment and the benefits of incorporating a multi-wave structure for effective temporal dynamic modeling. These results highlight the advantage of our multi-stage processing strategy over conventional U-Net architectures for the task at hand.

Theoretical Implications and Future Directions

SineNet introduces a new paradigm in neural PDE solving, focusing on addressing the temporal misalignment issue inherent in conventional architectures. Its success across different datasets suggests a promising direction for future research in leveraging both multi-scale and multi-stage processing for solving a broader range of PDEs. Furthermore, SineNet's adaptability in encoding various boundary conditions and potential for extensibility invites exploration into more complex systems and irregular domains.

Conclusion

SineNet represents a significant advancement in the design of neural PDE solvers, specifically tailored to address the challenges of time-dependent systems. By systematically reducing feature misalignment through a novel multi-stage architecture, SineNet sets a new standard in modeling dynamic systems governed by PDEs. This work not only demonstrates the potential of neural networks in scientific computing but also opens new avenues for future advancements in the field.

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