mLaSDI: Multi-stage latent space dynamics identification (2506.09207v2)

Abstract: Determining accurate numerical solutions of partial differential equations (PDEs) is an important task in many scientific disciplines. However, solvers can be computationally expensive, leading to the development of reduced-order models (ROMs). Recently, Latent Space Dynamics Identification (LaSDI) was proposed as a data-driven, non-intrusive ROM framework. LaSDI compresses the training data using an autoencoder and learns a system of user-chosen ordinary differential equations (ODEs), which govern the latent space dynamics. This allows for rapid predictions by interpolating and evolving the low-dimensional ODEs in the latent space. While LaSDI has produced effective ROMs for numerous problems, the autoencoder can have difficulty accurately reconstructing training data while also satisfying the imposed dynamics in the latent space, particularly in complex or high-frequency regimes. To address this, we propose multi-stage Latent Space Dynamics Identification (mLaSDI). With mLaSDI, several autoencoders are trained sequentially in stages, where each autoencoder learns to correct the error of the previous stages. We find that applying mLaSDI with small autoencoders results in lower prediction and reconstruction errors, while also reducing training time compared to LaSDI.

• The paper introduces a multi-stage ROM framework that sequentially trains autoencoders to correct residual errors and improve prediction accuracy.
• It integrates autoencoders for compression with SINDy for learning latent ODEs, achieving an order-of-magnitude reduction in error for high-frequency dynamics.
• Numerical tests on oscillating systems, unsteady wake flows, and the 1D-1V Vlasov equation demonstrate its potential for real-time simulation of complex phenomena.

Multi-stage Latent Space Dynamics Identification (mLaSDI)

The paper "mLaSDI: Multi-stage Latent Space Dynamics Identification" by Anderson et al. presents a novel framework for reduced-order modeling (ROM) that seeks to improve the accuracy and efficiency of predictions derived from complex systems governed by partial differential equations (PDEs). The authors address the inherent limitations associated with existing Latent Space Dynamics Identification (LaSDI) methods, particularly the challenges posed by data reconstruction and dynamics adherence in high-frequency regimes.

Overview and Contributions

Traditional methods, such as projection-based ROMs require knowledge of the governing equations, limiting their applicability in real-world scenarios where the system dynamics might be unknown or partially understood. Non-intrusive ROMs, such as LaSDI, circumvent this by establishing a framework where system dynamics are learned from data without the need to explicitly understand or incorporate the governing PDEs. The LaSDI framework employs autoencoders for data compression and Sparse Identification of Nonlinear Dynamics (SINDy) for learning the governing ordinary differential equations (ODEs) in the latent space.

The paper introduces a Multi-stage Latent Space Dynamics Identification (mLaSDI) framework, which enhances LaSDI's capabilities by sequentially training multiple autoencoders, with each stage focused on correcting the residuals left by the preceding ones. This allows mLaSDI to manage smaller autoencoder architectures, reducing both prediction errors and training times relative to LaSDI.

Strong Numerical Results

The paper demonstrates the efficacy of mLaSDI with three distinct numerical examples: a synthetic multiscale oscillating system, 2D unsteady wake flow, and the 1D-1V Vlasov equation. In each instance, mLaSDI showed substantial improvements in prediction accuracy over single-stage LaSDI implementations, achieving order-of-magnitude reductions in prediction error and efficiently capturing complex, high-frequency phenomena that were missed by traditional methods.

Of particular note is the performance in predicting the multiscale oscillating system, where mLaSDI reduced prediction errors from the LaSDI baseline of 10-20% to a range of 0.3-3%. Such results underscore the framework's capability to accommodate both low and high-frequency dynamics more effectively than existing methods.

Implications and Future Work

The paper's findings suggest substantial practical implications for scientific computing and engineering, notably in systems where PDEs are used to model complex phenomena such as fluid dynamics, material deformation, or plasma physics. The reduced computational load and improved accuracy make mLaSDI a promising candidate for integration into real-time simulation environments, where rapid predictions can be crucial.

The authors propose several avenues for future exploration in improving mLaSDI. These include integrating weak forms of SINDy for enhanced data representation and increasing autoencoder architecture complexity in subsequent stages to better handle high-frequency residuals. Moreover, the paper acknowledges the potential for overfitting due to improper scaling of training dynamics identification loss weights; thus, further investigations could focus on optimizing stage-wise network architectures to mitigate these risks.

Conclusion

Overall, the paper presents a significant advancement in non-intrusive ROM methodologies by demonstrating the potential of mLaSDI for high-fidelity modeling applications. The multi-stage approach not only mitigates common issues with data reconstruction in complex systems but also paves the way for broader application across various domains. As research within AI and computational modeling continues to expand, mLaSDI's contributions could serve as a cornerstone for developing faster and more accurate ROM frameworks in science and engineering.