Neural surrogates are widely used in scientific machine learning for fast prediction of three-dimensional (3D) thermo-fluid fields. However, generating training data using conventional numerical solvers often incurs substantial computational and storage costs. We propose to train an attention graph neural network by minimizing the finite-volume method (FVM) residuals of the governing equations. These residuals are evaluated directly on the mesh, requiring no labeled data. We evaluate the trained surrogates against computational fluid dynamics (CFD) references and a data-supervised baseline across four scenarios. On the two steady-state benchmarks, the FVM-loss model achieves an all-field normalized root-mean-square error (nRMSE) of 2.3-2.8%. It demonstrates close agreement with the CFD references, including the buoyancy-energy coupling. On the two parametric transient cases, the FVM-loss model outperforms the supervised baseline in terms of accuracy, while avoiding the data-generation cost entirely. These results indicate that the FVM loss can provide a practical training signal for neural surrogates and reduce the model development cost.
Translating unstructured clinical prescriptions into patient-specific foot orthoses (FOs) is hindered by a semantic-physical misalignment: high-level clinical intent is not mapped deterministically onto the 3D geometric parameters of the orthosis, and existing design workflows remain dependent on manual expertise with no instantaneous biomechanical validation. We present TANS-FO, a research prototype-a modular pipeline with closed-loop feedback for computational design automation of customized FOs, not a clinically validated therapeutic device. A Text-Aligned Neural Surrogate (TANS) uses cross-attention to project clinical-text embeddings onto a continuous lattice-density field, while a Graph Neural Network (GNN) surrogate predicts plantar stress in real time as a substitute for Finite Element Analysis (FEA). The framework is anchored on the open-access PicoFoot-5K anthropometric database (5,230 subjects; 30+ anatomical parameters). Under standardized quasi-static loading, the GNN surrogate agrees with an Abaqus reference solver (R^2 = 0.94), and the full pipeline synthesizes manufacturing-ready lattice insoles within minutes. On the Male 18-40 cohort, the proposed system attains a surrogate-predicted peak-pressure reduction of 34.7% over parametric CAD, with a fit error of 0.42 mm. Separately, an exploratory feasibility observation (n = 12; 2-week follow-up; no control group) using VAS pain reporting indicates short-term comfort improvement (VAS 6.4 -> 2.1), but this data is explicitly classified as preliminary observational evidence only-not evidence of clinical efficacy.
Neural surrogate models offer fast approximate mappings from PDE parameters to solutions, but they typically treat solving as a purely statistical task: once trained, they struggle to correct their own constraint violations and extrapolate beyond the training distribution. Recent hybrid methods promote physical correctness by targeting the PDE residual via gradient descent or Gauss--Newton steps, but inherit the compute cost and instability of the underlying classical optimizers. We show, theoretically and empirically, that numerically minimizing the PDE residual can be an unreliable proxy for reconstruction accuracy in ill-conditioned systems, explaining why these methods often do not make accurate predictions despite achieving low residuals. We propose error-conditioned Neural Solvers (ENS), built on a different principle: rather than an optimization target, the PDE residual field is passed as a direct input to the network at each iteration, enabling it to read the spatial structure of its own errors and learn an update policy to iteratively correct its predictions. Across four PDE families, ENS attains the highest prediction accuracy in the large majority of settings, with gains reaching $10\times$ on turbulent Kolmogorov flow, while avoiding the expensive compute cost of hybrid methods. ENS's learned correction policy generalizes under distribution shift, including zero-shot parameter changes and cross-equation transfer, where its relative advantage is largest in the ill-conditioned regimes where residual minimization is least reliable. Project website: https://neuralsolver.github.io/.
We propose latent-space diffusion posterior sampling (L-DPS), an approximate Bayesian framework for high-dimensional inverse problems governed by partial differential equations (PDEs). The method addresses three challenges in PDE-constrained inversion: implicit sample-based priors without tractable densities, high-dimensional spatially distributed parameters, and the high cost of repeated forward-model evaluations during posterior sampling. L-DPS combines a variational autoencoder, an unconditional latent diffusion model, diffusion posterior sampling, and a differentiable neural surrogate. The VAE maps the parameter field to a lower-dimensional latent space, the diffusion model learns an implicit prior score in this latent space, and DPS combines this learned prior with likelihood-based guidance. The likelihood gradient is evaluated through the decoder-surrogate composition, avoiding repeated calls to the full numerical PDE solver. We evaluate the method on an inverse Darcy flow problem with an unknown spatially distributed permeability field inferred from sparse and noisy pressure observations. L-DPS produces accurate and robust inverse solutions, reduces inference cost relative to full-space DPS, and outperforms amortized inverse baselines such as conditional latent diffusion and inverse FNO in sparse and noisy regimes. We further compare L-DPS with a KLE-MAP baseline and study mixed-prior generalization and the sensitivity of inversion accuracy to surrogate forward-model error.
High-fidelity simulations of free-surface flows using Lagrangian methods such as the Particle Finite Element Method (PFEM) are computationally demanding due to continuous domain updates and repeated solution of the governing equations. This challenge is further amplified by non-Newtonian rheologies, where material nonlinearities increase computational cost. These limitations motivate the development of efficient surrogate models to approximate PFEM dynamics at reduced cost. While data-driven deep learning approaches are promising, a key challenge is designing models that operate on arbitrary and evolving geometries. We propose a self-attention-based neural surrogate for PFEM simulations of free-surface flows. The architecture leverages attention mechanisms to model node interactions and capture complex spatial dependencies, while preserving the PFEM mesh discretization. This provides a geometric and topological framework for remeshing and node redistribution, maintaining high-quality spatial discretization during rollouts, improving long-term stability, and enabling reconstruction of derived mechanical quantities via standard finite element operators. Two attention formulations are considered: a standard self-attention mechanism and a linear variant that reduces computational cost and improves scalability. The models are evaluated on two- and three-dimensional free-surface flow benchmarks with evolving geometries, varying material parameters, and non-Newtonian fluids. Results show accurate prediction of transient dynamics and final configurations, with significantly improved scalability. The mesh-based formulation also enables direct reconstruction of quantities such as stress fields. Overall, the framework provides an accurate and scalable surrogate strategy for PFEM simulations in engineering-scale applications.
Cen Chen, Haitao Huang, Jiazhi Mao +3physics.app-ph cs.AI
Efficient exploration of the photonic crystal (PhC) lattice design space is essential for developing photonic crystal surface-emitting lasers. While coupled-wave theory (CWT) provides an effective physical framework, its computational cost remains prohibitive for large-scale exploration, driving the demand for neural surrogates. However, existing AI models underexploit two key factors of PhC unit-cell dielectric patterns indicated by CWT: spectral components and asymmetric structures, which largely govern devices' physical properties. This mismatch weakens surrogate accuracy and screening reliability, especially in structure-sensitive regions. To address this, we propose the Dual-Domain Symmetry-Aware Network (DDSNet). It integrates translation-equivariant spectral filtering with a symmetry-induced structural prior. The spectral filtering injects a spectral inductive bias into vision model while preserving translation equivariance on lattices. Meanwhile, the structural prior decomposes lattice features into irreducible representation-associated, symmetry-resolved components and processes them in separate branches. Experiments demonstrate that DDSNet significantly outperforms existing AI baselines in property prediction and high-throughput screening, exhibiting superior reliability in structure-sensitive regions. Crucially, component masking analyses reveal that the network successfully learns property-specific dependencies aligned with physical priors. These results indicate that DDSNet effectively captures physically meaningful structure-property relationships, establishing a highly reliable neural surrogate for PhC design space exploration.