Rishikesh Ranade, Chris Hill, Haiyang He, Amir Maleki, Jay Pathak
In this work we propose a hybrid solver to solve partial differential equation (PDE)s in the latent space. The solver uses an iterative inferencing strategy combined with solution initialization to improve generalization of PDE solutions. The solver is tested on an engineering case and the results show that it can generalize well to several PDE conditions.
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PDE-SHARP: PDE Solver Hybrids through Analysis and Refinement Passes
PDE-SHARP通过分析、生成与综合三阶段用廉价LLM推理替代昂贵科学计算，以少于13次评估实现PDE求解器精度平均提升4倍，并减少60-75%计算成本。
Enforcing governing equation constraints in neural PDE solvers via training-free projections
本文提出两种无训练投影方法，用于减少神经PDE求解器中约束方程的违反情况，提升求解精度。
TENG-BC: Unified Time-Evolving Natural Gradient for Neural PDE Solvers with General Boundary Conditions
TENG-BC通过统一时间演进自然梯度方法，实现对具有通用边界条件的神经PDE求解器的高精度求解，优于传统方法和PINN基线。
PDE-PFN: Prior-Data Fitted Neural PDE Solver
PDE-PFN是基于PFN架构的神经PDE求解器，通过在噪声近似解上预训练，实现跨异构方程泛化、噪声先验鲁棒性及零梯度更新推理，精度优于或相当基线。
Learning a Neural Solver for Parametric PDE to Enhance Physics-Informed Methods
本文提出学习参数偏微分方程的神经求解器，通过条件化梯度下降算法提升物理指导方法的优化效率和稳定性。
One of the main challenges in solving time-dependent partial differential equations is to develop computationally efficient solvers that are accurate and stable. Here, we introduce a graph neural network approach to finding efficient PDE solvers through learning using message-passing models. We first introduce domain invariant features for PDE-data inspired by classical PDE solvers for an efficient physical representation. Next, we use graphs to represent PDE-data on an unstructured mesh and show that message passing graph neural networks (MPGNN) can parameterize governing equations, and as a …
Tensor Gaussian Processes: Efficient Solvers for Nonlinear PDEs
本文提出基于张量高斯过程的求解器TGPS，通过一维高斯过程因子函数和张量分解降低计算复杂度，结合部分冻结策略和牛顿法提升效率，并在多个基准PDE上验证了其精度和效率优势。
OCL-PDE: A Generative Framework for PDE Inverse Problems with Observation-Complementary Latents
针对PDE逆问题的不适定性，提出OCL-PDE生成框架，利用观测互补潜变量结合观测重建未知场，基于物理感知AE和条件流匹配，提升重建精度与细节恢复。
Hamiltonian Neural PDE Solvers through Functional Approximation
本文提出HNS，将哈密顿泛函表示为神经场参数化的核积分，并用自动微分求泛函导数，从而在一维和二维PDE上提升稳定性、守恒性与长时泛化能力。
Adversarial Learning for Neural PDE Solvers with Sparse Data
本文针对数据稀缺下神经 PDE 求解器鲁棒性不足的问题，提出 SMART 通用学习策略，通过针对性改进模型薄弱环节降低泛化误差，并在多种 PDE 场景中显著提升预测精度。
Physics-informed Reduced Order Modeling of Time-dependent PDEs via Differentiable Solvers
本文提出物理信息降阶建模（$Φ$-ROM），通过将可微PDE求解器融入训练，使潜在动态符合物理规律，实现高精度泛化、长时预测和稀疏数据下的场重建，优于现有方法。
Physics-Informed Neural PDE Solvers via Spatio-Temporal MeanFlow
本文提出时空均流求解器，通过将生成速度场替换为物理PDE算子，实现高效预测，同时扩展均流约束到时空域，统一处理时变和静态PDE，实验表明其在精度和效率上优于基线方法。
Neural networks are increasingly being used to solve partial differential equations (PDEs), replacing slower numerical solvers. However, a critical issue is that neural PDE solvers require high-quality ground truth data, which usually must come from the very solvers they are designed to replace. Thus, we are presented with a proverbial chicken-and-egg problem. In this paper, we present a method, which can partially alleviate this problem, by improving neural PDE solver sample complexity -- Lie point symmetry data augmentation (LPSDA). In the context of PDEs, it turns out that we are able to qu…
MD-PNOP: Equation-Recast Neural Operators for Minimal-Data Extrapolation and PDE Solver Acceleration
本文提出MD-PNOP框架，通过将参数诱导的算子差转化为额外源项，实现单次训练配置到广泛未见参数设置的系统外推，同时嵌入迭代PDE求解器提升初始猜测，减少迭代次数且不牺牲精度。
M2NO: An Efficient Multi-Resolution Operator Framework for Dynamic Multi-Scale PDE Solvers
M2NO是一种高效的多分辨率运算框架，通过结合多网格结构和多小波空间，提升高维PDE求解的精度和效率，并在多种任务中表现优异。
Finding accurate solutions to partial differential equations (PDEs) is a crucial task in all scientific and engineering disciplines. It has recently been shown that machine learning methods can improve the solution accuracy by correcting for effects not captured by the discretized PDE. We target the problem of reducing numerical errors of iterative PDE solvers and compare different learning approaches for finding complex correction functions. We find that previously used learning approaches are significantly outperformed by methods that integrate the solver into the training loop and thereby a…
Same Target, Different Origins: Aligning Latent Rollout for Consistent PDE Emulation
本研究针对神经PDE求解器自回归展开的误差传播问题，提出ALR正则化方法，通过对齐潜在表示提升预测精度，减少不同展开起源的目标差异。
Numerical simulation of non-linear partial differential equations plays a crucial role in modeling physical science and engineering phenomena, such as weather, climate, and aerodynamics. Recent Machine Learning (ML) models trained on low-resolution spatio-temporal signals have shown new promises in capturing important dynamics in high-resolution signals, under the condition that the models can effectively recover the missing details. However, this study shows that significant information is often lost in the low-resolution down-sampled features. To address such issues, we propose a new approac…
Parameterized Representations via Implicit Stochastic Modulation for High-Dimensional and High-Order Neural PDE Solvers
提出PRISM框架，通过超生成器将物理参数映射为仿射调制器，实现参数化高维高阶随机神经PDE求解器，保持无偏性并降低内存消耗，支持零样本泛化。
Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers
研究针对神经偏微分方程求解器，提出能量流形自然梯度下降框架\EMNGD，将参数更新与函数空间能量曲率对齐，在黎曼流形上优化，证明相关性质，实验显示比现有基线精度更高、收敛更快，还能量化精度与成本权衡。
PDEFlow: Autonomous Agentic PDE Pipelines for Neural Operator Learning and Solver-Free Inference
提出PDEFlow自主智能体框架，将用户ODE/PDE描述转化为神经算子流水线，实现多轮自然语言输入处理、自动数据生成、算子训练与无求解器推理，大幅降低相关科学工程任务的手动干预成本。
Solver-Integrated Adversarial Attacking and Training of Neural Operators
本文从求解器角度研究神经算子的泛化与鲁棒性，提出求解器集成对抗攻击与训练方法，提升泛化与鲁棒性，实验表明深度集成更有效。
Flow Learners for PDEs: Toward a Physics-to-Physics Paradigm for Scientific Computing
本文提出流学习器（flow learners）范式，通过参数化传输向量场并积分生成轨迹，将PDE求解从状态预测转向物理上允许的未来传输建模，实现连续时间预测、不确定性量化及物理感知求解器设计。
Experimental Demonstration of an Optical Neural PDE Solver via On-Chip PINN Training
本文实验演示了一种光学神经PDE求解器，利用物理信息神经网络在光子芯片上进行免反向传播训练，为解决偏微分方程提供了新方法。
Toward Efficient Kernel-Based Solvers for Nonlinear PDEs
本文提出一种去除微分算子的核学习框架，通过标准核插值求导并结合网格配点与乘积核的 Kronecker 结构，避免构造完整 Gram 矩阵，从而高效求解非线性 PDE，并给出收敛性证明与数值验证。
Numerical PDE solvers outperform neural PDE solvers
本研究提出可微有限差分框架DeepFDM，将经典前向欧拉离散化嵌入卷积架构，在多类标量PDE基准测试中，误差、训练效率和参数量均显著优于主流神经PDE求解器，为参数化PDE的数据驱动求解与识别提供了稳健高效的透明基线。
Unisolver: PDE-Conditional Transformers Towards Universal Neural PDE Solvers
针对现有神经PDE求解器泛化性差的问题，提出以PDE组分为条件的Transformer模型Unisolver，结合物理结构与Transformer技术，在三个大规模基准上实现SOTA，迈向通用神经PDE求解器。
INC: An Indirect Neural Corrector for Auto-Regressive Hybrid PDE Solvers
针对自回归混合偏微分方程求解器的自回归误差问题，提出间接神经校正器INC，将校正整合到控制方程而非直接更新状态，降低误差放大并提升长期性能，在多案例中实现性能提升与加速，为高效物理仿真提供新方法。
GEPS: Boosting Generalization in Parametric PDE Neural Solvers through Adaptive Conditioning
GEPS通过自适应条件化与低秩快速适应机制，提升参数化PDE神经求解器对未见条件的泛化能力，并在时空预测任务中验证其有效性。
Over the last few decades, existing Partial Differential Equation (PDE) solvers have demonstrated a tremendous success in solving complex, non-linear PDEs. Although accurate, these PDE solvers are computationally costly. With the advances in Machine Learning (ML) technologies, there has been a significant increase in the research of using ML to solve PDEs. The goal of this work is to develop an ML-based PDE solver, that couples important characteristics of existing PDE solvers with ML technologies. The two solver characteristics that have been adopted in this work are: 1) the use of discretiza…
End-to-End Mesh Optimization of a Hybrid Deep Learning Black-Box PDE Solver
本研究探索了结合黑盒PDE求解器与深度图神经网络的混合模型在流体流动预测中的端到端训练可行性，通过零阶梯度估计器对求解器进行微分，在无需自动微分支持的情况下实现了优于冻结网格基线的性能，并通过预热启动加速收敛与提升泛化。
Constructing fast numerical solvers for partial differential equations (PDEs) is crucial for many scientific disciplines. A leading technique for solving large-scale PDEs is using multigrid methods. At the core of a multigrid solver is the prolongation matrix, which relates between different scales of the problem. This matrix is strongly problem-dependent, and its optimal construction is critical to the efficiency of the solver. In practice, however, devising multigrid algorithms for new problems often poses formidable challenges. In this paper we propose a framework for learning multigrid sol…
Pre-Generating Multi-Difficulty PDE Data for Few-Shot Neural PDE Solvers
通过预生成多难度的PDE数据，提升少样本神经PDE求解器的性能与效率。
We propose a novel multi-scale message passing neural network algorithm for learning the solutions of time-dependent PDEs. Our algorithm possesses both temporal and spatial multi-scale resolution features by incorporating multi-scale sequence models and graph gating modules in the encoder and processor, respectively. Benchmark numerical experiments are presented to demonstrate that the proposed algorithm outperforms baselines, particularly on a PDE with a range of spatial and temporal scales.
Autoregressive Renaissance in Neural PDE Solvers
本文解读ICLR 2022论文，介绍一种消息传递图神经网络，通过缓解自回归不稳定问题，在神经偏微分方程求解中达到或超越傅里叶神经算子与经典求解器的泛化性能。
Beyond Accuracy: EcoL2 Metric for Sustainable Neural PDE Solvers
针对现有神经PDE求解器只重精度、忽视计算碳排放的问题，提出平衡精度与全流程碳排放的EcoL2度量，经两类架构实验验证可实现性能与排放成本的整体评估，助力构建低环境影响的科学机器学习系统。
Latent PDE mapping for efficient physics-informed learning across geometries with limited data
研究提出潜在偏微分方程映射技术，用于有限数据下跨几何的物理信息学习。通过变形梯度处理特定几何的偏微分方程残差等，在解决阿利耶夫 - 潘菲洛夫偏微分方程时效果显著，能降低误差且训练计算成本适中，有助于创建可泛化模型。
Hybrid Iterative Solvers with Geometry-Aware Neural Preconditioners for Parametric PDEs
本文提出Geo-DeepONet，结合几何信息的神经预置条件器，提升混合迭代求解器在参数PDEs中的鲁棒性和效率。
CodePDE: An Inference Framework for LLM-driven PDE Solver Generation
CodePDE通过大语言模型生成PDE求解器，展示了LLM在复杂数学问题中的强大能力及潜在应用价值。
Singular parameters and missing limits in neural PDE solvers
本文针对神经PDE求解器中因参数无界增长导致的缺失极限问题，通过分析tanh网络并添加核导数来补全模型，使最佳逼近可达，并数值验证了其对优化的影响。
Warm-starting PDE solvers with any-dimensional machine learning
本文提出利用对称性条件，使低维训练的PDE求解器可零样本或热启动求解高维PDE，并在多个方程上验证了性能提升与数据效率优势。
Graph neural network (GNN) is a promising approach to learning and predicting physical phenomena described in boundary value problems, such as partial differential equations (PDEs) with boundary conditions. However, existing models inadequately treat boundary conditions essential for the reliable prediction of such problems. In addition, because of the locally connected nature of GNNs, it is difficult to accurately predict the state after a long time, where interaction between vertices tends to be global. We present our approach termed physics-embedded neural networks that considers boundary c…
Generative Latent Neural PDE Solver using Flow Matching
提出一种基于流匹配的潜在扩散模型，将PDE状态嵌入低维空间，降低计算成本，并在准确性和长期稳定性上优于确定性基线。
Active Learning for Neural PDE Solvers
针对神经PDE求解器训练数据成本高的问题，提出模块化主动学习基准AL4PDE，实验证明主动学习可大幅降低误差且生成的数据集可复用。
PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers
针对神经PDE求解器在长时间预测中因忽视高频信息而不稳定的问题，提出基于扩散模型的多步细化方法PDE-Refiner，在复杂流体动力学基准上实现更稳定准确的rollout，并提升数据效率与不确定性评估能力。
Physics-Aware Neural Implicit Solvers for multiscale, parametric PDEs with applications in heterogeneous media
提出物理感知神经隐式求解器（PANIS），利用加权残差虚拟数据与粗离散化隐式求解器，学习参数化PDE替代模型，适用于非均质材料及多尺度问题，支持不确定性量化。
Flexible and Efficient Probabilistic PDE Solvers through Gaussian Markov Random Fields
本文利用高斯过程先验的马尔可夫性质，通过随机偏微分方程离散化和稀疏线性代数大幅加速推理，使概率PDE求解器能够高效处理大规模非线性PDE，并支持更灵活的物理先验。
Error Analysis of Deep PDE Solvers for Option Pricing
本文通过Black--Scholes和Heston模型下的对比实验，系统分析了Deep Galerkin Method和TDGF两种深度PDE求解器在期权定价中的经验收敛速率与训练时间，为实际应用提供指导。
DGNN: A Neural PDE Solver Induced by Discontinuous Galerkin Methods
本文受IPDGM启发提出DGNN框架，以分段神经网络为试探空间、分段多项式为检验空间，在稳态与时变PDE数值实验中展现出高精度、训练高效及处理复杂情形的能力。
Latent Neural PDE Solver: a reduced-order modelling framework for partial differential equations
本文提出Latent Neural PDE Solver（LNS），通过非线性自编码器将PDE系统映射到粗离散潜空间并在其中进行时间演化预测，在单相与多相流等系统上实现了兼具精度与效率的降阶模拟。
Machine learned partial differential equation (PDE) solvers trade the reliability of standard numerical methods for potential gains in accuracy and/or speed. The only way for a solver to guarantee that it outputs the exact solution is to use a convergent method in the limit that the grid spacing $Δx$ and timestep $Δt$ approach zero. Machine learned solvers, which learn to update the solution at large $Δx$ and/or $Δt$, can never guarantee perfect accuracy. Some amount of error is inevitable, so the question becomes: how do we constrain machine learned solvers to give us the sorts of errors that…
Do Neural PDE Solvers Learn the Right Dynamics?
针对神经PDE求解器，提出评估框架，通过误差形成、集合几何和极端事件三方面检验其动力学保真度，发现预测精度提升不等于动力学保真度提升。
Error Analysis of Option Pricing via Deep PDE Solvers: Empirical Study
本研究通过实证对比和消融实验，系统分析了深度PDE求解器在高维期权定价中的三类误差来源，发现DBSDE方法性能优越且对期权规格变化鲁棒，并揭示了性能与批大小及时间步数的平方根成反比的关系，为实际应用提供了计算资源估算依据。
Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers
本文提出HINTS混合求解器，将DeepONet与松弛方法结合，利用谱偏差互补优势，实现PDE快速、可扩展且均匀收敛的数值求解。
Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators
本文提出统一视角，将物理感知神经网络和神经算子置于共享设计空间中，以揭示其在科学工作流中的关系和作用。
Efficient Neural PDE-Solvers using Quantization Aware Training
本文研究神经 PDE 求解器的推理效率问题，系统评估量化感知训练对权重和激活的压缩效果，证明其可在四个数据集、三种架构和三个 FLOPs 数量级上保持性能并实现成本-性能 Pareto 最优。
Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation
本文提出蒙特卡洛神经偏微分方程求解器（MCNP Solver），利用偏微分方程的概率表示以无监督方式训练神经求解器，将对流-扩散等方程的宏观现象建模为随机粒子集合，并通过Heun方法及邻近网格概率密度期望计算提升精度与效率，在多个方程上验证了其优越性。
PRISMA: Improving the Accuracy-Latency Frontier of Diffusion-based PDE Solvers Using Physics-Informed Spectral Attention
PRISMA通过将PDE残差作为频谱注意力机制整合到扩散神经算子中，实现了无需梯度下降的快速、稳健和准确的PDE求解。
LatentPINNs: Generative physics-informed neural networks via a latent representation learning
针对PINNs收敛慢且需针对不同PDE参数重新训练的问题，提出latentPINN框架，利用潜在扩散模型学习PDE参数分布的压缩潜在表示作为输入，通过两阶段训练，在Eikonal方程上验证了无需额外训练即可泛化到新速度模型。
UGrid: An Efficient-And-Rigorous Neural Multigrid Solver for Linear PDEs
本文提出融合 U-Net 与多重网格的线性 PDE 神经求解器 UGrid，兼具收敛与正确性证明、高精度及跨几何和 PDE 形式的泛化能力，并以残差损失实现稳定无监督训练。
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