PDEAgent-Bench: A Multi-Metric, Multi-Library Benchmark for PDE Solver Generation

PDE-to-solver code generation aims to automatically synthesize executable numerical solvers from partial differential equation (PDE) specifications. This task requires not only understanding the mathematical structure of PDEs, but also selecting appropriate discretization schemes and solver configurations, and correctly implementing the resulting formulations in finite-element method (FEM) libraries. Existing code generation benchmarks mainly evaluate syntactic correctness, or success on predefined test cases. To our knowledge, there is currently no publicly available benchmark specifically for PDE-to-solver code generation, and general-purpose code benchmarks do not fully capture the unique challenges of numerical PDE solution, such as ensuring solver accuracy, efficiency, and compatibility with professional FEM libraries. We introduce PDEAgent-Bench, to the best of our knowledge, the first multi-metric, multi-library benchmark for PDE-to-solver code generation. PDEAgent-Bench contains 645 instances across 6 mathematical categories and 11 PDE families, with common FEM libraries for DOLFINx, Firedrake, and deal.II. Each instance provides an agent-facing problem specification, a reference solution on a prescribed evaluation grid, and case-specific accuracy and runtime targets. PDEAgent-Bench adopts a staged evaluation framework in which generated solvers must sequentially pass executability, numerical accuracy, and computational efficiency checks. Experiments with representative LLMs and code agents show that models can often produce runnable code, but their pass rate drops substantially once accuracy and efficiency requirements are enforced. These results indicate that current agents remain limited in producing numerically reliable and efficient PDE solvers, and that PDEAgent-Bench provides a reproducible testbed grounded in the practical requirements of numerical PDE solving.

Learning Where to Embed: Noise-Aware Positional Embedding for Query Retrieval in Small-Object Detection

Transformer-based detectors have advanced small-object detection, but they often remain inefficient and vulnerable to background-induced query noise, which motivates deep decoders to refine low-quality queries. We present HELP (Heatmap-guided Embedding Learning Paradigm), a noise-aware positional-semantic fusion framework that studies where to embed positional information by selectively preserving positional encodings in foreground-salient regions while suppressing background clutter. Within HELP, we introduce Heatmap-guided Positional Embedding (HPE) as the core embedding mechanism and visualize it with a heatbar for interpretable diagnosis and fine-tuning. HPE is integrated into both the encoder and decoder: it guides noise-suppressed feature encoding by injecting heatmap-aware positional encoding, and it enables high-quality query retrieval by filtering background-dominant embeddings via a gradient-based mask filter before decoding. To address feature sparsity in complex small targets, we integrate Linear-Snake Convolution to enrich retrieval-relevant representations. The gradient-based heatmap supervision is used during training only, incurring no additional gradient computation at inference. As a result, our design reduces decoder layers from eight to three and achieves a 59.4% parameter reduction (66.3M vs. 163M) while maintaining consistent accuracy gains under a reduced compute budget across benchmarks. Code Repository: https://github.com/yidimopozhibai/Noise-Suppressed-Query-Retrieval