Text2PDE: Latent Diffusion Models for Accessible Physics Simulation

Recent advances in deep learning have inspired numerous works on data-driven solutions to partial differential equation (PDE) problems. These neural PDE solvers can often be much faster than their numerical counterparts; however, each presents its unique limitations and generally balances training cost, numerical accuracy, and ease of applicability to different problem setups. To address these limitations, we introduce several methods to apply latent diffusion models to physics simulation. Firstly, we introduce a mesh autoencoder to compress arbitrarily discretized PDE data, allowing for efficient diffusion training across various physics. Furthermore, we investigate full spatio-temporal solution generation to mitigate autoregressive error accumulation. Lastly, we investigate conditioning on initial physical quantities, as well as conditioning solely on a text prompt to introduce text2PDE generation. We show that language can be a compact, interpretable, and accurate modality for generating physics simulations, paving the way for more usable and accessible PDE solvers. Through experiments on both uniform and structured grids, we show that the proposed approach is competitive with current neural PDE solvers in both accuracy and efficiency, with promising scaling behavior up to $\sim$3 billion parameters. By introducing a scalable, accurate, and usable physics simulator, we hope to bring neural PDE solvers closer to practical use.

CaFA: Global Weather Forecasting with Factorized Attention on Sphere

Accurate weather forecasting is crucial in various sectors, impacting decision-making processes and societal events. Data-driven approaches based on machine learning models have recently emerged as a promising alternative to numerical weather prediction models given their potential to capture physics of different scales from historical data and the significantly lower computational cost during the prediction stage. Renowned for its state-of-the-art performance across diverse domains, the Transformer model has also gained popularity in machine learning weather prediction. Yet applying Transformer architectures to weather forecasting, particularly on a global scale is computationally challenging due to the quadratic complexity of attention and the quadratic increase in spatial points as resolution increases. In this work, we propose a factorized-attention-based model tailored for spherical geometries to mitigate this issue. More specifically, it utilizes multi-dimensional factorized kernels that convolve over different axes where the computational complexity of the kernel is only quadratic to the axial resolution instead of overall resolution. The deterministic forecasting accuracy of the proposed model on $1.5^\circ$ and 0-7 days' lead time is on par with state-of-the-art purely data-driven machine learning weather prediction models. We also showcase the proposed model holds great potential to push forward the Pareto front of accuracy-efficiency for Transformer weather models, where it can achieve better accuracy with less computational cost compared to Transformer based models with standard attention.

Latent Neural PDE Solver: a reduced-order modelling framework for partial differential equations

Neural networks have shown promising potential in accelerating the numerical simulation of systems governed by partial differential equations (PDEs). Different from many existing neural network surrogates operating on high-dimensional discretized fields, we propose to learn the dynamics of the system in the latent space with much coarser discretizations. In our proposed framework - Latent Neural PDE Solver (LNS), a non-linear autoencoder is first trained to project the full-order representation of the system onto the mesh-reduced space, then a temporal model is trained to predict the future state in this mesh-reduced space. This reduction process simplifies the training of the temporal model by greatly reducing the computational cost accompanying a fine discretization. We study the capability of the proposed framework and several other popular neural PDE solvers on various types of systems including single-phase and multi-phase flows along with varying system parameters. We showcase that it has competitive accuracy and efficiency compared to the neural PDE solver that operates on full-order space.

Inpainting Computational Fluid Dynamics with Deep Learning

Fluid data completion is a research problem with high potential benefit for both experimental and computational fluid dynamics. An effective fluid data completion method reduces the required number of sensors in a fluid dynamics experiment, and allows a coarser and more adaptive mesh for a Computational Fluid Dynamics (CFD) simulation. However, the ill-posed nature of the fluid data completion problem makes it prohibitively difficult to obtain a theoretical solution and presents high numerical uncertainty and instability for a data-driven approach (e.g., a neural network model). To address these challenges, we leverage recent advancements in computer vision, employing the vector quantization technique to map both complete and incomplete fluid data spaces onto discrete-valued lower-dimensional representations via a two-stage learning procedure. We demonstrated the effectiveness of our approach on Kolmogorov flow data (Reynolds number: 1000) occluded by masks of different size and arrangement. Experimental results show that our proposed model consistently outperforms benchmark models under different occlusion settings in terms of point-wise reconstruction accuracy as well as turbulent energy spectrum and vorticity distribution.

Pretraining Strategy for Neural Potentials

We propose a mask pretraining method for Graph Neural Networks (GNNs) to improve their performance on fitting potential energy surfaces, particularly in water systems. GNNs are pretrained by recovering spatial information related to masked-out atoms from molecules, then transferred and finetuned on atomic forcefields. Through such pretraining, GNNs learn meaningful prior about structural and underlying physical information of molecule systems that are useful for downstream tasks. From comprehensive experiments and ablation studies, we show that the proposed method improves the accuracy and convergence speed compared to GNNs trained from scratch or using other pretraining techniques such as denoising. On the other hand, our pretraining method is suitable for both energy-centric and force-centric GNNs. This approach showcases its potential to enhance the performance and data efficiency of GNNs in fitting molecular force fields.

HNO: Hyena Neural Operator for solving PDEs

Numerically solving partial differential equations (PDEs) typically requires fine discretization to resolve necessary spatiotemporal scales, which can be computationally expensive. Recent advances in deep learning have provided a new approach to solving PDEs that involves the use of neural operators. Neural operators are neural network architectures that learn mappings between function spaces and have the capability to solve partial differential equations based on data. This study utilizes a novel neural operator called Hyena, which employs a long convolutional filter that is parameterized by a multilayer perceptron. The Hyena operator is an operation that enjoys sub-quadratic complexity and state space model to parameterize long convolution that enjoys global receptive field. This mechanism enhances the model's comprehension of the input's context and enables data-dependent weight for different PDE instances. To measure how effective the layers are in solving PDEs, we conduct experiments on Burger's equation and Navier Stokes equation. Our findings indicate Hyena Neural operator can serve as an efficient and accurate model for learning PDEs' solution operator. The data and code used can be found at: https://github.com/Saupatil07/Hyena-Neural-Operator

Scalable Transformer for PDE Surrogate Modeling

Transformer has shown state-of-the-art performance on various applications and has recently emerged as a promising tool for surrogate modeling of partial differential equations (PDEs). Despite the introduction of linear-complexity variant, applying attention to a large number of grid points can result in instability and is still expensive to compute. In this work, we propose Factorized Transformer(FactFormer), which is based on an axial factorized kernel integral. Concretely, we introduce a learnable projection operator that decomposes the input function into multiple sub-functions with one-dimensional domain. These sub-functions are then evaluated and used to compute the instance-based kernel with an axial factorized scheme. We showcase that the proposed model is able to simulate 2D Kolmogorov flow on a 256 by 256 grid and 3D smoke buoyancy on a 64 by 64 by 64 grid with good accuracy and efficiency. In addition, we find out that with the factorization scheme, the attention matrices enjoy a more compact spectrum than full softmax-free attention matrices.

Physics Informed Token Transformer

Solving Partial Differential Equations (PDEs) is the core of many fields of science and engineering. While classical approaches are often prohibitively slow, machine learning models often fail to incorporate complete system information. Over the past few years, transformers have had a significant impact on the field of Artificial Intelligence and have seen increased usage in PDE applications. However, despite their success, transformers currently lack integration with physics and reasoning. This study aims to address this issue by introducing PITT: Physics Informed Token Transformer. The purpose of PITT is to incorporate the knowledge of physics by embedding partial differential equations (PDEs) into the learning process. PITT uses an equation tokenization method to learn an analytically-driven numerical update operator. By tokenizing PDEs and embedding partial derivatives, the transformer models become aware of the underlying knowledge behind physical processes. To demonstrate this, PITT is tested on challenging PDE neural operators in both 1D and 2D prediction tasks. The results show that PITT outperforms the popular Fourier Neural Operator and has the ability to extract physically relevant information from governing equations.

Denoise Pre-training on Non-equilibrium Molecules for Accurate and Transferable Neural Potentials

Machine learning methods, particularly recent advances in equivariant graph neural networks (GNNs), have been investigated as surrogate models to expensive ab initio quantum mechanics (QM) approaches for molecular potential predictions. However, building accurate and transferable potential models using GNNs remains challenging, as the quality and quantity of data are greatly limited by QM calculations, especially for large and complex molecular systems. In this work, we propose denoise pre-training on non-equilibrium molecular conformations to achieve more accurate and transferable GNN potential predictions. Specifically, GNNs are pre-trained by predicting the random noises added to atomic coordinates of sampled non-equilibrium conformations. Rigorous experiments on multiple benchmarks reveal that pre-training significantly improves the accuracy of neural potentials. Furthermore, we show that the proposed pre-training approach is model-agnostic, as it improves the performance of different invariant and equivariant GNNs. Notably, our models pre-trained on small molecules demonstrate remarkable transferability, improving performance when fine-tuned on diverse molecular systems, including different elements, charged molecules, biomolecules, and larger systems. These results highlight the potential for leveraging denoise pre-training approaches to build more generalizable neural potentials for complex molecular systems.

A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction

Machine learning models are gaining increasing popularity in the domain of fluid dynamics for their potential to accelerate the production of high-fidelity computational fluid dynamics data. However, many recently proposed machine learning models for high-fidelity data reconstruction require low-fidelity data for model training. Such requirement restrains the application performance of these models, since their data reconstruction accuracy would drop significantly if the low-fidelity input data used in model test has a large deviation from the training data. To overcome this restraint, we propose a diffusion model which only uses high-fidelity data at training. With different configurations, our model is able to reconstruct high-fidelity data from either a regular low-fidelity sample or a sparsely measured sample, and is also able to gain an accuracy increase by using physics-informed conditioning information from a known partial differential equation when that is available. Experimental results demonstrate that our model can produce accurate reconstruction results for 2d turbulent flows based on different input sources without retraining.