Model Reduction and Neural Networks

Neueste Paper

• Discrete energy as an exact label-free training objective for finite-element surrogates

  Ruifeng Cao (The University of Manchester), Xidan Song (Wuhan University) · 7. August 2026

  Supervised training of finite-element (FE) surrogate models requires reference solutions, and each reference solution is obtained by solving the system that the surrogate is intended to replace. The assembled discrete potential energy provides a training signal that requires no reference solution. T…

• Alternating Levenberg-Marquardt Training of Physics-Informed Neural Networks with Fourier-Enhanced Features

  Yulun Wu, Matthieu Barreau, Miguel Aguiar, Karl H. Johansson · 7. August 2026

  Physics-informed neural networks (PINNs) often fail to accurately resolve partial differential equations (PDEs) with high-frequency or multi-scale solutions, as well as strongly nonlinear problems. Two factors underlie this difficulty: spectral bias, the tendency of neural networks to underfit high-…

• Kastor: An efficient fine-tuning strategy for generative emulation of PDE simulations

  Guillaume Couairon, Alexis Jacq, Yu-Han Wu, Renu Singh, Yana Hasson, Quentin Berthet, Romuald Elie · 7. August 2026

  Machine learning offers a promising avenue to accelerate physical simulations by replacing computationally expensive traditional Partial Differential Equation (PDE) solvers with fast, differentiable surrogate models. However, standard auto-regressive ML emulators often suffer from error accumulation…

• From Non-Convex Self-Concordant Regularization to Scalable Quasi-Newton Training of PINNs

  Chenhao Si, Kang An, Shiqian Ma, Ming Yan · 6. August 2026

  Physics-informed neural networks (PINNs) often require high-accuracy quasi-Newton refinement to obtain reliable partial differential equation solutions, but their residual objectives can exhibit indefinite, nearly singular, and poorly scaled local curvature. Regularized quasi-Newton methods provide …

• Improving Auto-Design of Neural PDE Solvers with a Domain-Specific Language

  Shengxin Kong, Liwen Xu, Jingwen Fu · 6. August 2026

  Neural PDE solver auto-design is fundamentally a search-space representation problem. In the space of unrestricted Python programs, valid solvers form an extremely sparse subset: most candidate programs are syntactically incorrect, semantically incompatible, or numerically unstable. Direct code gene…

• Towards Understanding Gradient Flow Dynamics of Homogeneous Neural Networks Beyond the Origin

  Akshay Kumar, Jarvis Haupt · 6. August 2026

  Recent works exploring the training dynamics of homogeneous neural network weights under gradient flow with small initialization have established that in the early stages of training, the weights remain small and near the origin, but converge in direction. Building on this, the current paper studies…

• Continual-Learning Physics-Informed Neural Networks for Parameterized Partial Differential Equations

  Xujia Chen, Xinyue Hu, Letian Chen, Yi Liu, Wenhui Fan · 6. August 2026

  Physics-informed neural networks (PINNs) incorporate governing equations into neural-network training and can approximate PDE solutions without requiring large observational datasets. Parameterized PINNs (ParamPINNs) further take physical parameters as inputs, allowing a single model to represent a …

• A Physics-Informed Hybrid Neural Operator for Transient Magnetization Prediction in Power Magnetics

  Yachao Zhu, Qiujie Huang, Sinan Li, Yang Li, Gang Lei, Jianguo Zhu · 5. August 2026

  Magnetic components in high-frequency, high-power-density converters are increasingly driven by non-sinusoidal flux-density waveforms with fast transitions, minor-loop operation, dc bias, and temperature variation. Under these conditions, steady-state core-loss formulas and single-valued material cu…

• Verifier-Guided Model Discovery for Physical Dynamical Systems with Pretrained Symbolic Transformers

  Farbod Faraji, Francesco Belardinelli · 5. August 2026

  Reliable forecasting of nonlinear physical systems underpins scientific discovery and engineering decision-making. Yet high-fidelity simulations are prohibitively costly, and machine-learning surrogates can be opaque and encode assumptions about system dynamics, limiting generalizability. Pretrained…

• Modeling Unknown Nonlocal PDE Systems via Flow Map Learning

  Zhongshu Xu, Ying Li, Yanzhi Zhang, Dongbin Xiu · 4. August 2026

  Nonlocal partial differential equations arise in many applications but are often difficult to model and learn because of the presence of nonlocal operators. We present a flow-map learning (FML) framework for modeling unknown nonlocal PDEs directly from solution data. Rather than learning or approxim…

• Quantitative Sobolev Approximation Bounds for Neural Operators with Empirical Validation on Burgers Equation

  Nicole Hao · 4. August 2026

  Neural operators have emerged as a powerful tool for learning mappings between infinite-dimensional function spaces. However, their approximation properties in Sobolev norms remain poorly quantified, even though these norms control both function values and derivatives and are the natural metrics for…

• Convex Neural Energy Elements: Monolithic Finite-Element Assembly of Geometry-Parameterized Neural Operators with Stability and Error Guarantees

  Hongyue Jiang, Jianjiang Zhan, Chenzhuo Zhang, Fan Wang · 4. August 2026

  Extending the neural-operator element method from individually trained, fixed-geometry neural elements to a library of reusable, geometry-parameterized element types fails structurally: a field-predicting operator trained by value regression induces an energy whose assembled Hessian is indefinite, a…

• LieStoNet: Learning Lie Symmetries from Spatiotemporal Data for Stochastic Dynamical Systems

  Shida Liu, Abhishek Gupta, Sumit Sinha, L. Mahadevan · 4. August 2026

  Symmetry is central to modern machine learning and physics: invariances and equivariances improve sample efficiency, robustness, and out-of-distribution generalization, while symmetry principles guide scientific modeling. Yet for stochastic dynamical systems the relevant continuous symmetries are ra…

• CoSynFlow: Conformal Symplectic Neural Flows for Cross-System Prediction of Dissipative Hamiltonian Dynamics

  Baige Xu, Takaharu Yaguchi · 4. August 2026

  Learning solution operators for differential equations is a central problem in scientific machine learning. However, many neural operator methods optimize prediction accuracy without explicitly enforcing the geometric structure of the dynamics. Structure-preserving models such as SympNets and Symple…

• HyperODE: Zero-Shot Surrogate for Simulation and Inference of Dynamical Systems

  Ajitesh Srivastava · 4. August 2026

  Understanding and controlling complex dynamical systems often requires executing thousands of numerical simulations across vast parametric landscapes, which is time-consuming. Machine learning surrogates significantly accelerate simulation by predicting state trajectories across different initializa…

• Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditions

  Ziang Chen, Liqiang Huang · 4. August 2026

  We study the complexity of approximating high-dimensional second-order elliptic PDEs with homogeneous boundary conditions on the unit hypercube using Barron spaces. Under suitable Barron assumptions on the coefficients and forcing term, we prove that the solutions can be approximated to any prescrib…

• An Embedded RISC-V Evaluation of Kolmogorov--Arnold Networks in Hard-Constrained Recurrent Physics-Informed Models

  Enzo Nicolas Spotorno, Josafat Leal Filho · 4. August 2026

  Hard-constrained recurrent physics-informed networks (HRPINNs) embed known dynamics inside a recurrent numerical integrator and restrict a neural branch to learning only the residual dynamics that the first-principles model does not capture. Kolmogorov--Arnold Networks (KANs) have been proposed as p…

• Hybrid Lagrangian-Eulerian Model for Lagrangian Fluid Simulation

  Ruoyan Li, Wei Wang, Yizhou Sun · 4. August 2026

  Pure Lagrangian neural simulators offer geometric flexibility and exact advection, making them well-suited for modeling moving domains and free surfaces. However, the absence of a fixed global reference frame introduces two severe limitations: a spatial bottleneck, in which model capacity is wasted …

• Dynamics-aware identification of governing equations from sparse and noisy data

  Pongpisit Thanasutives, Yoshinobu Kawahara · 3. August 2026

  Sparse identification of nonlinear dynamics (SINDy) and PDE functional identification (PDE-FIND) recover parsimonious ordinary and partial differential equations (ODEs and PDEs) from data. However, sparse and noisy temporal measurements can make derivative estimates unreliable. To address this probl…

• DFSC: Error-Controlled Differentiable Mittag-Leffler Propagation for Fractional Scientific Machine Learning

  Ning Hu, Haitao Duan, Shuqun Li, Chuyang Hu · 3. August 2026

  Fractional scientific machine learning requires numerical operators that can be differentiated, batched, accelerated, and composed with neural networks. When the dominant linear fractional evolution is known through a Mittag-Leffler propagator, repeatedly reconstructing that response with a history …

• Feature Interaction Modeling for Physics-Informed Neural Networks and Neural Operators

  Quan Gu, Hongxia Liu · 3. August 2026

  This work embeds feature interaction modules derived from factorization machines (FMs) into physics-informed neural networks (PINNs) and neural operator learning, to enhance model expressiveness for solution manifolds of parameterized partial differential equations (PDEs). Motivated by the second-or…

• Freeze, Then Select: Structured Field Adapters and Stability-Validated Weak Selection for PDE Discovery from Sparse Observations

  Juncheng Zhong, Chenghuang Shen, Jianfeng Liu, Zhengdong Xiao, Longjiu Luo, Qianrong Wang, Wenjun Xu, Wenlian Lu · 3. August 2026

  PDE discovery from sparse observations requires reconstructing a continuous field and selecting the correct differential terms. Our analysis of optimization paths in coupled neural PDE discovery reveals three behaviors: the exact support can persist to the end of training, appear only transiently, o…

• In-situ Autoguidance: Eliciting Self-Correction in Diffusion Models

  Enhao Gu, Haolin Hou · 3. August 2026

  The generation of high-quality, diverse, and prompt-aligned images is a central goal in image-generating diffusion models. The popular classifier-free guidance (CFG) approach improves quality and alignment at the cost of reduced variation, creating an inherent entanglement of these effects. Recent w…

• HERO: History-Enriched Rollout Training for Long-Horizon Autoregressive Neural Operators

  Jiaquan Zhang, Shuxu Chen, Haifan Meng, Yi Lu, Zhihan Lyu, Fan Mo, Wei Dong, Yang Yang, Chaoning Zhang · 3. August 2026

  Neural operators provide fast surrogates for time-dependent partial differential equations (PDEs) by applying a learned evolution operator recursively to its own predictions, but this autoregressive rollout feeds every prediction error back as input, so local errors accumulate. Existing rollout-trai…

• PIKS: Universal Physics-Informed Kernel Methods

  Joachim Bona-Pellissier, Giacomo Meanti, Matteo Santacesaria, Lorenzo Rosasco · 30. Juli 2026

  Physics-informed machine learning incorporates physical principles --often expressed via differential operators-- into data-driven models. While physics-informed neural networks (PINNs) dominate empirical applications, the complexity of neural network architectures and optimization landscapes hinder…