Simulating the time evolution of Partial Differential Equations (PDEs) of large-scale systems is crucial in many scientific and engineering domains such as fluid dynamics, weather forecasting and their inverse optimization problems. However, both classical solvers and recent deep learning-based surrogate models are typically extremely computationally intensive, because of their local evolution: they need to update the state of each discretized cell at each time step during inference. Here we develop Latent Evolution of PDEs (LE-PDE), a simple, fast and scalable method to accelerate the simulation and inverse optimization of PDEs. LE-PDE learns a compact, global representation of the system and efficiently evolves it fully in the latent space with learned latent evolution models. LE-PDE achieves speedup by having a much smaller latent dimension to update during long rollout as compared to updating in the input space. We introduce new learning objectives to effectively learn such latent dynamics to ensure long-term stability. We further introduce techniques for speeding-up inverse optimization of boundary conditions for PDEs via backpropagation through time in latent space, and an annealing technique to address the non-differentiability and sparse interaction of boundary conditions. We test our method in a 1D benchmark of nonlinear PDEs, 2D Navier-Stokes flows into turbulent phase and an inverse optimization of boundary conditions in 2D Navier-Stokes flow. Compared to state-of-the-art deep learning-based surrogate models and other strong baselines, we demonstrate up to 128x reduction in the dimensions to update, and up to 15x improvement in speed, while achieving competitive accuracy.
more »
« less
This content will become publicly available on May 1, 2024
Learning Controllable Adaptive Simulation for Multi-resolution Physics
Simulating the time evolution of physical systems is pivotal in many scientific and engineering problems. An open challenge in simulating such systems is their multi-resolution dynamics: a small fraction of the system is extremely dynamic, and requires very fine-grained resolution, while a majority of the system is changing slowly and can be modeled by coarser spatial scales. Typical learning-based surrogate models use a uniform spatial scale, which needs to resolve to the finest required scale and can waste a huge compute to achieve required accuracy. We introduced Learning controllable Adaptive simulation for Multiresolution Physics (LAMP) as the first full deep learning-based surrogate model that jointly learns the evolution model and optimizes appropriate spatial resolutions that devote more compute to the highly dynamic regions. LAMP consists of a Graph Neural Network (GNN) for learning the forward evolution, and a GNNbased actor-critic for learning the policy of spatial refinement and coarsening. We introduced learning techniques that optimize LAMP with weighted sum of error and computational cost as objective, allowing LAMP to adapt to varying relative importance of error vs. computation tradeoff at inference time. We evaluated our method in a 1D benchmark of nonlinear PDEs and a challenging 2D mesh-based simulation. We demonstrated that our LAMP outperforms state-of-the-art deep learning surrogate models, and can adaptively trade-off computation to improve long-term prediction error: it achieves an average of 33.7% error reduction for 1D nonlinear PDEs, and outperforms MeshGraphNets + classical Adaptive Mesh Refinement (AMR) in 2D mesh-based simulations.
more »
« less
- NSF-PAR ID:
- 10471866
- Publisher / Repository:
- NSF-PAR
- Date Published:
- Journal Name:
- International Conference on Learning Representations (ICLR)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
more » « less
Abstract Surrogate models driven by sizeable datasets and scientific machine-learning methods have emerged as an attractive microstructure simulation tool with the potential to deliver predictive microstructure evolution dynamics with huge savings in computational costs. Taking 2D and 3D grain growth simulations as an example, we present a completely overhauled computational framework based on graph neural networks with not only excellent agreement to both the ground truth phase-field methods and theoretical predictions, but enhanced accuracy and efficiency compared to previous works based on convolutional neural networks. These improvements can be attributed to the graph representation, both improved predictive power and a more flexible data structure amenable to adaptive mesh refinement. As the simulated microstructures coarsen, our method can adaptively adopt remeshed grids and larger timesteps to achieve further speedup. The data-to-model pipeline with training procedures together with the source codes are provided.
-
Today's scientific simulations require a significant reduction of data volume because of extremely large amounts of data they produce and the limited I/O bandwidth and storage space. Error-bounded lossy compression has been considered one of the most effective solutions to the above problem. However, little work has been done to improve error-bounded lossy compression for Adaptive Mesh Refinement (AMR) simulation data. Unlike the previous work that only leverages 1D compression, in this work, we propose to leverage high-dimensional (e.g., 3D) compression for each refinement level of AMR data. To remove the data redundancy across different levels, we propose three pre-process strategies and adaptively use them based on the data characteristics. Experiments on seven AMR datasets from a real-world large-scale AMR simulation demonstrate that our proposed approach can improve the compression ratio by up to 3.3X under the same data distortion, compared to the state-of-the-art method. In addition, we leverage the flexibility of our approach to tune the error bound for each level, which achieves much lower data distortion on two application-specific metrics.more » « less
-
Flying snakes are the only snakes on Earth capable of aerial gliding, taking advantage of fluid dynamic principles to leap from point to point among the trees. During their gliding, the locomotion of aerial undulation is observed. We hypothesize that this locomotion and its associated unsteady vortex dynamics are critical to their aerodynamic performance. However, there is a lack of detailed three-dimensional flow field information around the snake body in gliding due to the difficulties in experimental flow visualizations of live animals. In this study, a computation fluid dynamics (CFD) study has been conducted to study the fluid dynamics of a snake-like gliding. A mathematical equation describing the horizontal undulation motion was applied for constructing snake-like 3D computational models and a series of flow simulations were conducted. An immersed-boundary-method (IBM)-based direct numerical simulation (DNS) flow solver along with adaptive mesh refinement (AMR) was used in the simulation. Specifically, different head positions, corresponding to different horizontal wave shapes and their effect on aerodynamic performance, flow field and wake structures behind the body will be studied. In addition, the dynamic undulating motion is introduced in the model and a CFD simulation is also conducted. Results from this study are expected to bring a step stone to understanding snake-inspired locomotion.more » « less
-
An accurate understanding of omnidirectional environment lighting is crucial for high-quality virtual object rendering in mobile augmented reality (AR). In particular, to support reflective rendering, existing methods have leveraged deep learning models to estimate or have used physical light probes to capture physical lighting, typically represented in the form of an environment map. However, these methods often fail to provide visually coherent details or require additional setups. For example, the commercial framework ARKit uses a convolutional neural network that can generate realistic environment maps; however the corresponding reflective rendering might not match the physical environments. In this work, we present the design and implementation of a lighting reconstruction framework called LITAR that enables realistic and visually-coherent rendering. LITAR addresses several challenges of supporting lighting information for mobile AR. First, to address the spatial variance problem, LITAR uses two-field lighting reconstruction to divide the lighting reconstruction task into the spatial variance-aware near-field reconstruction and the directional-aware far-field reconstruction. The corresponding environment map allows reflective rendering with correct color tones. Second, LITAR uses two noise-tolerant data capturing policies to ensure data quality, namely guided bootstrapped movement and motion-based automatic capturing. Third, to handle the mismatch between the mobile computation capability and the high computation requirement of lighting reconstruction, LITAR employs two novel real-time environment map rendering techniques called multi-resolution projection and anchor extrapolation. These two techniques effectively remove the need of time-consuming mesh reconstruction while maintaining visual quality. Lastly, LITAR provides several knobs to facilitate mobile AR application developers making quality and performance trade-offs in lighting reconstruction. We evaluated the performance of LITAR using a small-scale testbed experiment and a controlled simulation. Our testbed-based evaluation shows that LITAR achieves more visually coherent rendering effects than ARKit. Our design of multi-resolution projection significantly reduces the time of point cloud projection from about 3 seconds to 14.6 milliseconds. Our simulation shows that LITAR, on average, achieves up to 44.1% higher PSNR value than a recent work Xihe on two complex objects with physically-based materials.more » « less
