arXiv Machine Learning
Jun 2
Flowers: A Warp Drive for Neural PDE Solvers
arXiv:2603. 04430v2 Announce Type: replace Abstract: We introduce Flowers, a neural architecture for learning PDE solution operators built entirely from multihead warps.
Unsupervised Adaptation of PDE Foundation Models
arXiv:2608. 07053v1 Announce Type: new Abstract: Pretrained partial differential equation (PDE) foundation models can generalize across different equations, but adapting them to unseen PDE systems typically requires dense solution data, which is often expensive or unavailable.
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arXiv Machine Learning
Sep 17
HiLNO: A Hierarchical Latent Neural Operator with Multi-Scale Supervision for PDEs on General Geometries
HiLNO is a hierarchical latent neural operator that builds a fine‑to‑coarse‑to‑fine latent space and incorporates multi‑scale supervision and anisotropic Gaussian attention to preserve spatial information in PDE solutions with multiscale structures. The hierarchy reduces information loss during compression, while multi‑scale supervision aligns intermediate predictions with downsampled targets, and anisotropic attention facilitates feature transfer across scales. Experiments on standard PDE benchmarks and a large‑scale automotive aerodynamics task show that HiLNO achieves competitive accuracy while cutting parameter count by 84.4% and FLOPs by 69.2% compared with LinearNO, and it generalizes effectively to unseen spatial resolutions.
arXiv Machine Learning
2d ago
Residuals Are Not Enough: Limits of Physics-Informed Pre-Training for Scientific Foundation Models
arXiv:2503.19081v2 Announce Type: replace Abstract: Scientific foundation models (SciFMs) aim to learn generalizable representations of physical systems governed by partial differential equations (PD...
arXiv Machine Learning
Jun 5
When Attention Beats Fourier: Multi-Scale Transformers for PDE Solving on Irregular Domains
arXiv:2605. 08318v2 Announce Type: replace Abstract: We study the problem of \emph{architecture selection} for deep learning models trained to solve partial differential equations (PDEs), asking when transformer-based architectures with learned attention outperform Fourier-domain neural operators.
arXiv Machine Learning
Jul 27
Latent PDE mapping for efficient physics-informed learning across geometries with limited data
arXiv:2607. 22215v1 Announce Type: new Abstract: In this study, we introduce latent PDE mapping, a broadly applicable physics-informed learning technique designed to enable efficient geometric generalization with sparse training data.
arXiv Statistics ML
3d ago
Warm-starting PDE solvers with any-dimensional machine learning
arXiv:2609.38916v1 Announce Type: new Abstract: Any-dimensional machine learning models, such as graph neural networks (GNNs), can be naturally trained and evaluated on inputs of different sizes and...
arXiv Machine Learning
Sep 7
Disentangling Attention in Deep Operator Learning: A Controlled Study of Data-Driven and Physics-Informed Architectures
The paper investigates how different attention mechanisms affect the performance of DeepONet neural operators. Five variants—varying in cross‑attention, self‑attention, tokenization, and attention depth—are trained in both data‑driven and physics‑informed settings on one‑ and two‑dimensional PDE benchmarks. Results show that per‑sensor tokenization with cross‑attention consistently reduces error, while branch self‑attention helps only in complex spatial problems, and deeper cross‑attention yields diminishing returns with higher cost.
arXiv Machine Learning
Aug 14
Physics-Informed Laplace Neural Operator for Solving Partial Differential Equations
arXiv:2602. 12706v2 Announce Type: replace Abstract: Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs).
arXiv Machine Learning
Sep 15
Single-condition neural solvers encode transferable response spaces for parametric differential equations
The paper demonstrates that a neural solver trained on a single condition can generate a reusable response space via its output Jacobian, enabling efficient cross‑condition solution transfer. By introducing Linearized Subspace Transfer (LST) and Active Transfer Modeling (ATM), the authors recover target solutions through residual minimization and selectively acquire additional response spaces based on coverage indicators. Experiments on six PDE systems show that this approach reduces error and offline construction cost compared to physics‑informed baselines, achieving significant accuracy gains and rapid target adaptation.
arXiv AI
Aug 25
One-Step Evolution for Long-Time Extrapolation: An Error-Bound-Informed and Prior-Guided Neural Residual Framework for Autonomous PDEs
arXiv:2608.22026v1 Announce Type: new Abstract: Accurate simulation of the long-time evolution of systems governed by partial differential equations (PDEs) is central to scientific computing. Among e...
arXiv Machine Learning
Aug 26
Physics-Integrated Operator Learning via Gaussian Splatting Representations
arXiv:2608.24049v1 Announce Type: new Abstract: Neural operators provide efficient surrogates for spatiotemporal PDE systems, but purely data-driven formulations often accumulate substantial errors d...
arXiv AI
Aug 5
PRISMA: Improving the Accuracy-Latency Frontier of Diffusion-based PDE Solvers Using Physics-Informed Spectral Attention
arXiv:2512. 01370v2 Announce Type: replace-cross Abstract: Diffusion-based solvers for partial differential equations (PDEs) are often bottle-necked by slow gradient-based test-time optimization routines that use PDE residuals for loss guidance.