Spatio-temporal, multi-field deep learning of shock propagation in meso-structured media

The ability to predict how shock waves traverse porous and architected materials is a decisive factor in planetary defense, national security, and the race to achieve inertial fusion energy. Yet capturing pore collapse, anomalous Hugoniot responses, and localized heating -- phenomena that can determine the success of asteroid deflection or fusion ignition -- has remained a major challenge despite recent advances in single-field and reduced representations. We introduce a multi-field spatio-temporal deep learning model (MSTM) that unifies seven coupled fields -- pressure, density, temperature, energy, material distribution, and two velocity components -- into a single autoregressive surrogate. Trained on high-fidelity hydrocode data, MSTM runs about a thousand times faster than direct simulation, achieving errors below 4\% in porous materials and below 10\% in lattice structures. Unlike prior single-field or operator-based surrogates, MSTM resolves sharp shock fronts while preserving integrated quantities such as mass-averaged pressure and temperature to within 5\%. This advance transforms problems once considered intractable into tractable design studies, establishing a practical framework for optimizing meso-structured materials in planetary impact mitigation, inertial fusion energy, and national security.

Data-Driven Autoencoder Numerical Solver with Uncertainty Quantification for Fast Physical Simulations

Traditional partial differential equation (PDE) solvers can be computationally expensive, which motivates the development of faster methods, such as reduced-order-models (ROMs). We present GPLaSDI, a hybrid deep-learning and Bayesian ROM. GPLaSDI trains an autoencoder on full-order-model (FOM) data and simultaneously learns simpler equations governing the latent space. These equations are interpolated with Gaussian Processes, allowing for uncertainty quantification and active learning, even with limited access to the FOM solver. Our framework is able to achieve up to 100,000 times speed-up and less than 7% relative error on fluid mechanics problems.

GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencoder

Numerically solving partial differential equations (PDEs) can be challenging and computationally expensive. This has led to the development of reduced-order models (ROMs) that are accurate but faster than full order models (FOMs). Recently, machine learning advances have enabled the creation of non-linear projection methods, such as Latent Space Dynamics Identification (LaSDI). LaSDI maps full-order PDE solutions to a latent space using autoencoders and learns the system of ODEs governing the latent space dynamics. By interpolating and solving the ODE system in the reduced latent space, fast and accurate ROM predictions can be made by feeding the predicted latent space dynamics into the decoder. In this paper, we introduce GPLaSDI, a novel LaSDI-based framework that relies on Gaussian process (GP) for latent space ODE interpolations. Using GPs offers two significant advantages. First, it enables the quantification of uncertainty over the ROM predictions. Second, leveraging this prediction uncertainty allows for efficient adaptive training through a greedy selection of additional training data points. This approach does not require prior knowledge of the underlying PDEs. Consequently, GPLaSDI is inherently non-intrusive and can be applied to problems without a known PDE or its residual. We demonstrate the effectiveness of our approach on the Burgers equation, Vlasov equation for plasma physics, and a rising thermal bubble problem. Our proposed method achieves between 200 and 100,000 times speed-up, with up to 7% relative error.

Certified data-driven physics-informed greedy auto-encoder simulator

A parametric adaptive greedy Latent Space Dynamics Identification (gLaSDI) framework is developed for accurate, efficient, and certified data-driven physics-informed greedy auto-encoder simulators of high-dimensional nonlinear dynamical systems. In the proposed framework, an auto-encoder and dynamics identification models are trained interactively to discover intrinsic and simple latent-space dynamics. To effectively explore the parameter space for optimal model performance, an adaptive greedy sampling algorithm integrated with a physics-informed error indicator is introduced to search for optimal training samples on the fly, outperforming the conventional predefined uniform sampling. Further, an efficient k-nearest neighbor convex interpolation scheme is employed to exploit local latent-space dynamics for improved predictability. Numerical results demonstrate that the proposed method achieves 121 to 2,658x speed-up with 1 to 5% relative errors for radial advection and 2D Burgers dynamical problems.