Deep models have recently emerged as promising tools to solve partial differential equations (PDEs), known as neural PDE solvers. While neural solvers trained from either simulation data or physics-informed loss can solve PDEs reasonably well, they are mainly restricted to a few instances of PDEs, e.g. a certain equation with a limited set of coefficients. This limits their generalization to diverse PDEs, preventing them from being practical surrogate models of numerical solvers. In this paper, we present Unisolver, a novel Transformer model trained on diverse data and conditioned on diverse PDEs, aiming towards a universal neural PDE solver capable of solving a wide scope of PDEs. Instead of purely scaling up data and parameters, Unisolver stems from the theoretical analysis of the PDE-solving process. Inspired by the mathematical structure of PDEs that a PDE solution is fundamentally governed by a series of PDE components such as equation symbols and boundary conditions, we define a complete set of PDE components and flexibly embed them as domain-wise and point-wise deep conditions for Transformer PDE solvers. Integrating physical insights with recent Transformer advances, Unisolver achieves consistent state-of-the-art on three challenging large-scale benchmarks, showing impressive performance and generalizability. Code is available at [https://github.com/thuml/Unisolver](https://github.com/thuml/Unisolver).

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.

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.

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.

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.