Yuezhou Ma

Lanxiang Xing, Haixu Wu, Yuezhou Ma et al.

Fluid prediction is a long-standing challenge due to the intrinsic high-dimensional non-linear dynamics. Previous methods usually utilize the non-linear modeling capability of deep models to directly estimate velocity fields for future prediction. However, skipping over inherent physical properties but directly learning superficial velocity fields will overwhelm the model from generating precise or physics-reliable results. In this paper, we propose the HelmFluid toward an accurate and interpretable predictor for fluid. Inspired by the Helmholtz theorem, we design a HelmDynamics block to learn Helmholtz dynamics, which decomposes fluid dynamics into more solvable curl-free and divergence-free parts, physically corresponding to potential and stream functions of fluid. By embedding the HelmDynamics block into a Multiscale Multihead Integral Architecture, HelmFluid can integrate learned Helmholtz dynamics along temporal dimension in multiple spatial scales to yield future fluid. Compared with previous velocity estimating methods, HelmFluid is faithfully derived from Helmholtz theorem and ravels out complex fluid dynamics with physically interpretable evidence. Experimentally, HelmFluid achieves consistent state-of-the-art in both numerical simulated and real-world observed benchmarks, even for scenarios with complex boundaries.

Haixu Wu, Huakun Luo, Yuezhou Ma et al.

Physics-informed neural networks (PINNs) have been widely applied to solve partial differential equations (PDEs) by enforcing outputs and gradients of deep models to satisfy target equations. Due to the limitation of numerical computation, PINNs are conventionally optimized on finite selected points. However, since PDEs are usually defined on continuous domains, solely optimizing models on scattered points may be insufficient to obtain an accurate solution for the whole domain. To mitigate this inherent deficiency of the default scatter-point optimization, this paper proposes and theoretically studies a new training paradigm as region optimization. Concretely, we propose to extend the optimization process of PINNs from isolated points to their continuous neighborhood regions, which can theoretically decrease the generalization error, especially for hidden high-order constraints of PDEs. A practical training algorithm, Region Optimized PINN (RoPINN), is seamlessly derived from this new paradigm, which is implemented by a straightforward but effective Monte Carlo sampling method. By calibrating the sampling process into trust regions, RoPINN finely balances optimization and generalization error. Experimentally, RoPINN consistently boosts the performance of diverse PINNs on a wide range of PDEs without extra backpropagation or gradient calculation. Code is available at this repository: https://github.com/thuml/RoPINN.

Haixu Wu, Minghao Guo, Yuezhou Ma et al.

Attention with bias, which extends standard attention by introducing prior knowledge as an additive bias matrix to the query-key scores, has been widely deployed in vision, language, protein-folding and other advanced scientific models, underscoring its status as a key evolution of this foundational module. However, introducing bias terms creates a severe efficiency bottleneck in attention computation. It disrupts the tightly fused memory-compute pipeline that underlies the speed of accelerators like FlashAttention, thereby stripping away most of their performance gains and leaving biased attention computationally expensive. Surprisingly, despite its common usage, targeted efficiency optimization for attention with bias remains absent, which seriously hinders its application in complex tasks. Diving into the computation of FlashAttention, we prove that its optimal efficiency is determined by the rank of the attention weight matrix. Inspired by this theoretical result, this paper presents FlashBias based on the low-rank compressed sensing theory, which can provide fast-exact computation for many widely used attention biases and a fast-accurate approximation for biases in general formalizations. FlashBias can fully take advantage of the extremely optimized matrix multiplication operation in modern GPUs, achieving 1.5$\times$ speedup for Pairformer in AlphaFold 3, and over 2$\times$ speedup for attention with bias in vision and language models without loss of accuracy. Code is available at this repository: https://github.com/thuml/FlashBias.

Yuezhou Ma, Haixu Wu, Hang Zhou et al.

Physics sensing plays a central role in many scientific and engineering domains, which inherently involves two coupled tasks: reconstructing dense physical fields from sparse observations and optimizing scattered sensor placements to observe maximum information. While deep learning has made rapid advances in sparse-data reconstruction, existing methods generally omit optimization of sensor placements, leaving the mutual enhancement between reconstruction and placement on the shelf. To change this suboptimal practice, we propose PhySense, a synergistic two-stage framework that learns to jointly reconstruct physical fields and to optimize sensor placements, both aiming for accurate physics sensing. The first stage involves a flow-based generative model enhanced by cross-attention to adaptively fuse sparse observations. Leveraging the reconstruction feedback, the second stage performs sensor placement via projected gradient descent to satisfy spatial constraints. We further prove that the learning objectives of the two stages are consistent with classical variance-minimization principles, providing theoretical guarantees. Extensive experiments across three challenging benchmarks, especially a 3D geometry dataset, indicate PhySense achieves state-of-the-art physics sensing accuracy and discovers informative sensor placements previously unconsidered. Code is available at this repository: https://github.com/thuml/PhySense.

Haixu Wu, Yuezhou Ma, Hang Zhou et al.

Physics-informed neural networks (PINNs) have earned high expectations in solving partial differential equations (PDEs), but their optimization usually faces thorny challenges due to the unique derivative-dependent loss function. By analyzing the loss distribution, previous research observed the propagation failure phenomenon of PINNs, intuitively described as the correct supervision for model outputs cannot ''propagate'' from initial states or boundaries to the interior domain. Going beyond intuitive understanding, this paper provides a formal and in-depth study of propagation failure and its root cause. Based on a detailed comparison with classical finite element methods, we ascribe the failure to the conventional single-point-processing architecture of PINNs and further prove that propagation failure is essentially caused by the lower gradient correlation of PINN models on nearby collocation points. Compared to superficial loss maps, this new perspective provides a more precise quantitative criterion to identify where and why PINN fails. The theoretical finding also inspires us to present a new PINN architecture, named ProPINN, which can effectively unite the gradients of region points for better propagation. ProPINN can reliably resolve PINN failure modes and significantly surpass advanced Transformer-based models with 46% relative promotion.

Yuxuan Wang, Haixu Wu, Yuezhou Ma et al.

Deep time series forecasting has emerged as a booming direction in recent years. Despite the exponential growth of community interests, researchers are sometimes confused about the direction of their efforts due to minor improvements on standard benchmarks. In this paper, we notice that, unlike image recognition, whose well-acknowledged and realizable goal is 100% accuracy, time series forecasting inherently faces a non-zero error lower bound due to its partially observable and uncertain nature. To pinpoint the research objective and release researchers from saturated tasks, this paper focuses on a fundamental question: how to estimate the performance upper bound of deep time series forecasting? Going beyond classical series-wise predictability metrics, e.g., ADF test, we realize that the forecasting performance is highly related to window-wise properties because of the sequence-to-sequence forecasting paradigm of deep time series models. Based on rigorous statistical tests of over 2,800 newly trained deep forecasters, we discover a significant exponential relationship between the minimum forecasting error of deep models and the complexity of window-wise series patterns, which is termed the accuracy law. The proposed accuracy law successfully guides us to identify saturated tasks from widely used benchmarks and derives an effective training strategy for large time series models, offering valuable insights for future research.

Hang Zhou, Haixu Wu, Haonan Shangguan et al.

Deep learning has emerged as a transformative tool for the neural surrogate modeling of partial differential equations (PDEs), known as neural PDE solvers. However, scaling these solvers to industrial-scale geometries with over $10^8$ cells remains a fundamental challenge due to the prohibitive memory complexity of processing high-resolution meshes. We present Transolver-3, a new member of the Transolver family as a highly scalable framework designed for high-fidelity physics simulations. To bridge the gap between limited GPU capacity and the resolution requirements of complex engineering tasks, we introduce two key architectural optimizations: faster slice and deslice by exploiting matrix multiplication associative property and geometry slice tiling to partition the computation of physical states. Combined with an amortized training strategy by learning on random subsets of original high-resolution meshes and a physical state caching technique during inference, Transolver-3 enables high-fidelity field prediction on industrial-scale meshes. Extensive experiments demonstrate that Transolver-3 is capable of handling meshes with over 160 million cells, achieving impressive performance across three challenging simulation benchmarks, including aircraft and automotive design tasks.