A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder (2009.11990v2)

Abstract: Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simulations, in which the intrinsic solution space falls into a subspace with a small dimension, i.e., the solution space has a small Kolmogorov n-width. However, for physical phenomena not of this type, e.g., any advection-dominated flow phenomena, such as in traffic flow, atmospheric flows, and air flow over vehicles, a low-dimensional linear subspace poorly approximates the solution. To address cases such as these, we have developed a fast and accurate physics-informed neural network ROM, namely nonlinear manifold ROM (NM-ROM), which can better approximate high-fidelity model solutions with a smaller latent space dimension than the LS-ROMs. Our method takes advantage of the existing numerical methods that are used to solve the corresponding full order models. The efficiency is achieved by developing a hyper-reduction technique in the context of the NM-ROM. Numerical results show that neural networks can learn a more efficient latent space representation on advection-dominated data from 1D and 2D Burgers' equations. A speedup of up to 2.6 for 1D Burgers' and a speedup of 11.7 for 2D Burgers' equations are achieved with an appropriate treatment of the nonlinear terms through a hyper-reduction technique. Finally, a posteriori error bounds for the NM-ROMs are derived that take account of the hyper-reduced operators.

• The paper introduces a nonlinear manifold solution representation via a shallow masked autoencoder that outperforms traditional LS-ROMs in capturing complex dynamics.
• The paper employs an efficient hyper-reduction technique that achieves notable speedups—up to 11.7x in 2D Burgers’ equations—without sacrificing accuracy.
• The paper provides rigorous a posteriori error bounds, reinforcing the model’s theoretical credibility and practical utility in high-fidelity simulations.

A Fast and Accurate Physics-Informed Neural Network Reduced Order Model with Shallow Masked Autoencoder: Overview and Implications

This paper introduces an innovative approach to model reduction for solving complex physical simulations, addressing inefficiencies of traditional linear subspace reduced order models (LS-ROMs). LS-ROMs, while effective in scenarios where the solution space has a small Kolmogorov $n$-width, often fall short in accurately approximating physical phenomena characterized by advection-dominated flows, such as atmospheric or vehicular airflows. To surmount these limitations, the authors propose a nonlinear manifold reduced order model (NM-ROM), leveraging the representational power of physics-informed neural networks.

Key Contributions

• Nonlinear Manifold Solution Representation: The cornerstone of the proposed method is the nonlinear manifold solution representation, formed through a shallow masked autoencoder. This neural network model, trained on solution data from full order model (FOM) simulations, surpasses the LS-ROMs in effectively capturing complex dynamics with a reduced dimension of the latent space.
• Efficient Hyper-Reduction Technique: A significant advancement in the proposed NM-ROM is the hyper-reduction technique, which optimizes the computational efficiency by focusing the model's capacity on essential features. Notably, the paper presents a hyper-reduction strategy tailored to the NM-ROM context, facilitating superior speed without compromising accuracy.
• Numerical Validation on Nonlinear Dynamics: The authors validate the NM-ROM through numerical experiments on 1D and 2D Burgers' equations. The results showcase substantial improvements in terms of speed and accuracy as compared to LS-ROMs. The NM-ROM achieves up to 2.6x speedup for 1D Burgers' and an impressive 11.7x speedup for 2D scenarios, demonstrating its robustness and effectiveness in handling advection-dominated problems.
• A Posteriori Error Bounds: The paper derives a posteriori error bounds for the proposed NM-ROMs, accounting for the hyper-reduced operators and lending theoretical credibility to the methodology.

Theoretical and Practical Implications

The introduction of NM-ROM represents a significant contribution to the modeling of complex physical systems. The key theoretical implication lies in demonstrating that replacing the linear subspace representation with a nonlinear manifold can vastly enhance model accuracy and efficiency for advection-dominated problems. Practically, this advancement suggests that the NM-ROM could be particularly beneficial in fields requiring rapid decision-making based on high-fidelity simulations, such as aerodynamics, climate modeling, and real-time system controls.

Outlook on Future AI Development

This research opens avenues for future developments in AI-driven reduced order modeling. The use of physics-informed neural networks highlights the potential for intelligent systems to learn efficient representations of complex dynamics. Future work could explore the scalability of NM-ROMs to even higher-dimensional systems and broader categories of nonlinear dynamics. Furthermore, integrating such models with real-time data assimilation frameworks could enhance the predictive capabilities of simulation models, potentially revolutionizing applications in real-time monitoring and control systems.

The results and methodologies presented in this paper provide a promising framework for advancing the efficacy of model reduction techniques, offering a path toward more integrated and intelligent simulation tools. The balance between theoretical rigor and practical application illustrates the potential for significant advances in computational modeling spurred by AI techniques.