numerical-methods

FlowBudget 提出一种免训练的采样预算估计方法，通过仿真自由的插值探针估计求解器相对的采样复杂度，从而在不进行 FID-vs-NFE 扫描的情况下确定流匹配模型所需的函数求值次数（NFE）。

论文提出 D-SLR（Disjoint Row-Sparse plus Low-Rank）分解，作为截断 SVD 的闭式替代方案，通过要求每行要么原样存储、要么由低秩拟合来消除耦合，在平方误差下不损失最优性，且无需迭代求解或调参。

Eli Bendersky's blog post explains Monte Carlo simulations through simple, hands-on examples, including estimating pi by dart-throwing and simulating the card game SET to answer combinatorial probability questions.

LinearSolveBench is a new benchmark for evaluating linear solvers in C, focusing on speed, accuracy, and generality for large sparse systems to promote algorithmic advances.

These lecture notes introduce Physics Informed Neural Networks (PINNs) and Neural Operators, covering their implementation in PyTorch and open-source libraries like NVIDIA PhysicsNeMo, with applications in engineering and physics.

The article investigates why the Gauss-Seidel method, which converges faster theoretically, performs slower than Jacobi due to loop-carried dependencies, and shows how loop unrolling can improve performance.

This paper proposes a Physics-Informed Error Field Learning (PIEFL) framework for Physics-Informed Neural Networks, introducing an auxiliary error network to improve solution accuracy in solving partial differential equations under computational constraints.

This paper introduces Neural Networks with Local Converging Inputs (NNLCI) to accelerate PDE-based option pricing, combining coarse and refined mesh solutions to reduce RMSE by 4–12x in multi-asset Black–Scholes and Heston models.

A technical blog post introducing Walk on Decomposed Subdomains (WODS), a grid-free Monte Carlo method for solving elliptic PDEs with complex geometries, contrasted with finite differences and finite elements.

AutoPDE is a code agent that explicitly represents solver strategies for partial differential equations, improving pass rate by 14.2% over baselines on the PDE Agent Bench.

Blog post surveys fast hyperbolic tangent approximations—Taylor, Padé, splines, and bit-level tricks—for neural-network and real-time audio use.