NeurIPS 2026 Papers

The Physics-Informed Neural Network (PINN) literature has produced a rapidly expanding catalogue of methods. We divide a PINN training pipeline systematically into four components and observe that many existing benchmarks study these components in isolation. These components are domain-importance adaptation, loss balancing, the neural network architecture, and the optimization scheme. Moreover, the studies set the surrounding and not considered components of the pipeline to a minimal default that is no longer representative of current best practice approaches. Conclusions drawn under such conditions lead to suboptimal results. In this work, we propose a new evaluation protocol that (1) treats the four components as an integrated system; (2) requires as a key aspect a competitive implementation of the components not currently under investigation; (3) specifies the metrics and reporting format, and (4) is accompanied by structured documentation that ensures the traceability of evaluations across studies. Based on this comprehensive protocol, we release with PINNeval an innovative JAX package containing 18 different methods from the current literature as well as nine efficiently implemented benchmark problems. Using PINNeval, we illustrate our evaluation protocol in relation to established benchmark studies and report on results that challenge established assumptions.

We introduce PINNBench, a benchmark and evaluation study for training-policy selection in hybrid PINN-operator PDE solvers, contributed to the NeurIPS 2026 Evaluations and Datasets (ED) Track. PINNBench comprises 1,539 recorded runs and probes across 13 PDE configurations drawn from 8 equation families, 3–5 training policies per PDE, and 10 seeds per condition, with seven standardized metrics, Wilcoxon signed-rank tests, paired bootstrap 95% confidence intervals, Cohen's d, and Benjamini–Hochberg FDR correction. Baseline evaluation of 5 routing methods establishes three findings that bear directly on how PDE solvers should be evaluated. (i) No single policy dominates: hybrid_full wins 7/13, hybrid_no_gate 4/13, pinn_only 2/13. (ii) Accuracy and regret dissociate: under relative L2, a physics-loss heuristic and a learned Random Forest router reach identical 62% top-1 accuracy, but the heuristic suffers mean regret 0.0850 versus 0.0011 — a 77x gap. Evaluation by accuracy alone is misleading for PDE policy routing. (iii) Causal-loss effectiveness is stage-mediated: 7/9 PDEs improve with pretraining (+5.9% mean) while 5/9 degrade without it (-4.2%). A new leave-one-PDE-family-out evaluation shows that the strongest family-level generalizer uses no PDE-identity features, addressing the concern that PDE-aware routing memorizes PDE identity. All 1,539 result JSONs, PDE implementations, policy configurations, statistical pipeline, and routing infrastructure are released as open-source artifacts.

Protein--nucleic-acid recognition underlies aptamer discovery and nucleic-acid therapeutics, yet computational models often trade mechanistic interpretability for predictive flexibility. Hybrid interpretable--latent architectures risk fallback takeover, where the latent branch silently becomes the true predictor while the explicit pathway remains decorative. We introduce PIGRAM, a patch--motif grammar model that decomposes proteins into local residue patches and nucleic-acid partners into secondary-structure motifs, learning a sign-separated grammar over explicit physicochemical and geometric attributes to preserve rare favorable interactions. Heterogeneous rule families are selectively refined into a coarse-to-fine atlas, and a frozen-grammar residual Transformer provides sparse, bounded corrections without displacing the grammar as the primary predictor. On a strict sequence-pair-disjoint benchmark, PIGRAM achieves a Pearson correlation of 0.494, approaching the unconstrained Transformer (0.511) while exposing signed rules, fine subtypes, and route-gated corrections for each prediction. The learned rules recover bidirectional patch--motif mechanisms, and residual corrections concentrate selectively on grammar-hard regions. In fluorescence ELISA experiments, suppressive-rule relief improved GFP binding, whereas favorable-rule disruption weakened NELF binding, providing direct experimental support for grammar-guided design. Together, these results show that interpretable grammar can serve as both a predictive scaffold and an experimentally actionable design principle for aptamer optimization.

Full-space electron density generation is fundamental to modeling molecular quantum states, ground-state properties, and chemical interactions. DFT provides a principled route to electron density, but high-fidelity calculations require costly self-consistent iterations, limiting large-scale quantum-chemical modeling. Deep learning offers a promising surrogate, yet existing methods often rely on indirect density-related prior, including coordinate queries, predefined basis functions, orbital coefficients, or spherical atomic density grids. These prior introduce reconstruction overhead and can hinder continuous full-space modeling across sharp nuclear cores and sparse vacuum regions. Here, we introduce PI-EDG, a physics-informed voxel-space framework that maps molecular geometries directly to full-space electron-density fields by combining quantum-mechanical prior injection, hybrid FNO-CNN modeling of local and non-local interactions and magnitude-aware DS-LMoE routing for extreme density ranges. On EDBench, PI-EDG achieves an MAE of 3.405$\times$10$^{-3}$, a total electron-number MAE of 0.472, and an average inference time of 133.4 ms for one molecule. Compared with graph-based, basis-based, and voxel-based baselines, PI-EDG demonstrates better predictive accuracy, improved robustness across extreme density regimes, and better physical consistency while maintaining efficient full-space generation. These results suggest that PI-EDG offers a new, direct, and efficient route for accelerating DFT-based electron density calculations.

Optical microscopy lets us watch living bacteria in real time without stains or labels, and it remains the standard instrument for identifying species across clinical, environmental and industrial microbiology. Yet real samples rarely hold a single species. They arrive as mixtures and sometimes carry an organism no model has ever seen. What happens when a model trained on familiar mixtures meets one it has never encountered? No benchmark has asked this of phase-contrast images, so we built one. **PHOEBI** brings together 120,000 phase-contrast images of 40 mixtures of six bacterial species that we cultured and imaged ourselves, and it holds whole mixtures out of training so that every test is a combination the model has never seen. The answer is sobering. Classifiers that are nearly perfect on the mixtures they know collapse on new ones, and no choice of backbone or pretraining rescues them. The features turn out to be sound and the failure lies in how the evidence within each image is pooled. A culture looks the same everywhere, and decoders built on this simple fact hold steady where large networks fail even though one of them is never trained by gradient descent. The same features go further still. They notice when a species outside the catalogue appears, and the simplest way of giving it a class of its own proves the most effective while leaving the known species almost untouched. PHOEBI turns the mixed culture, the everyday case at the bench, into a measurable target for open-world recognition.

Large pretrained models have achieved remarkable success across diverse domains, yet adapting them to multiple downstream tasks remains challenging: full fine-tuning is costly, while maintaining separate task-specific models is storage-inefficient and fails to produce a unified multi-task model. Low-Rank Adaptation (LoRA) offers an efficient alternative for Model Merging by representing each task adaptation as a low-rank update over frozen pretrained weights. However, existing LoRA merging methods often treat task-specific updates as directly comparable objects and merge them in their original low-rank parameter spaces. This overlooks two key factors: which update directions provide reliable shared support across tasks, and how these directions should be re-parameterized with respect to the pretrained weights before merging. To address this limitation, we propose Pretrained-Guided Shared Basis (PGSB), a two-stage framework for LoRA model merging. PGSB first identifies shared directions among task-specific LoRA updates, and then re-parameterizes these directions using their interaction with the pretrained weights, producing a more suitable basis for merging. Extensive experiments demonstrate that PGSB consistently outperforms state-of-the-art baselines across multiple benchmarks.

3D Gaussian Splatting (3DGS) has become an established representation for real-time novel view synthesis. However, preserving fine geometric structures and appearance details typically requires millions of Gaussian primitives, resulting in substantial storage and transmission overhead. 3DGS compression has been extensively studied through attribute quantization, entropy coding, and lightweight reparameterization. However, existing methods often operate on a fixed backbone and therefore do not explicitly model the joint redundancy between geometry and appearance in the original dense Gaussian set. To address this limitation, we propose \textbf{Pyramidal Gaussian Mixture Splatting} (PGMS), a plug-and-play 3DGS compression framework that reformulates dense Gaussian primitives into a rate-controllable pyramidal representation with scale-dependent attributes. Specifically, PGMS first constructs a mixture pyramid by progressively partitioning dense Gaussians into hierarchical mixture centers, where different levels perform attribute-specific clustering to form shared prototypes. It then employs a rate-adaptive capacity allocator to determine the tiered budget under a target compression ratio. To further improve rendering quality, we introduce a compositing-aware parameter update rule that incorporates information from dense Gaussians into pyramid primitives according to their image-space contributions. Experiments on standard benchmarks show that our method delivers strong compression with high rendering fidelity and consistently outperforms state-of-the-art 3DGS compression methods.

Extending feed-forward 3D Gaussian Splatting (3DGS) to large-scale scenes remains challenging. Existing global reconstruction methods incur prohibitive computational overhead on long sequences; while streaming-based online methods are more efficient, their inherent forgetting behavior and unidirectional dependence still limit reconstruction quality. To address this, we propose PF-SGS, a pose-free streaming feed-forward 3DGS for large-scale scene reconstruction. PF-SGS leverages a persistent hidden state as scene memory to integrate streaming inputs frame-by-frame, progressively recovering scene geometry and camera poses in a self-supervised manner. At its core lie two targeted designs: Adaptive State Gating, which suppresses long-term memory degradation by dynamically regulating update magnitudes to ensure stable state evolution; and Delayed Gaussian Modeling, which introduces bounded look-ahead observations to compensate for insufficient local geometric constraints under streaming inputs, thereby improving the geometric consistency. Experiments demonstrate that PF-SGS processes sequences of varying lengths with linear time complexity, achieving quality comparable to pose-prior-based large-scale reconstruction baselines.

AdaBoost sequentially fits so-called weak learners to minimize an exponential loss, which penalizes misclassified data points more severely than other loss functions like cross-entropy. Paradoxically, AdaBoost generalizes well in practice as the number of weak learners grows. In the present work, we introduce Penalized Exponential Loss (PENEX), a new formulation of the multi-class exponential loss that is theoretically grounded and, in contrast to the existing formulation, amenable to optimization via first-order methods, making it a practical objective for training neural networks. We demonstrate that PENEX effectively increases margins of data points, which can be translated into a generalization bound. Empirically, across computer vision and language tasks, PENEX improves neural network generalization in low-data regimes, matching and in some settings outperforming established regularizers at comparable computational cost. Our results highlight the potential of the exponential loss beyond its application in AdaBoost.

Diffusion image generators owe much of their practical success to accurate denoising networks and strong numerical samplers. Yet the two components are not fully matched. During sampling, the denoiser is asked to make predictions based on imperfect noisy states formed by model predictions that differ from the training noisy states formed by real data. This train-test gap, known otherwise as exposure bias in literature, caps image generation quality (i.e., blurry, lacking in details or unnaturally-shaped). To mitigate the problem, we propose PEEK, a training-free deterministic sampler plugin. At each eligible denoising step, PEEK first performs a provisional one-step update, evaluates the denoiser once at the look-ahead state, and either keeps the current clean prediction or replaces it with this one-step-later prediction to form the actual noisy state. By taking the additional step, we generally obtain a higher-signal, more-image-like prediction for the noisy state, mitigating the exposure bias. To optimize the replacement positions, PEEK searches for a binary replacement schedule with a greedy metric-driven search made efficient by the PEEK prefix and baseline noise-schedule-rebasing caches; for a 9-step search, the caches reduce the one-candidate worst-case vanilla denoiser-evaluation count by 69.3\%. Across pixel-space and latent-space diffusion models with unconditional, label-guided, and text-guided settings on CIFAR-10, CelebA-HQ, LSUN Bedroom, ImageNet, and MS-COCO datasets, PEEK improves over NFE-matched DPM-Solver++ and DEIS baselines, with five-run mean FID reductions up to 39.3\% under the same deployment cost. Code will be released.

Large language model (LLM) agents increasingly operate over long and recurring external contexts, like document corpora and code repositories. Across invocations, existing approaches preserve either the agent's trajectory, passive access to raw material, or task-level strategies. None of them preserves what we argue is most needed for repeated same-context workloads: reusable orientation knowledge (e.g., what the context contains, how it is organized, and which entities, constants, and schemas have historically been useful) about the recurring context itself. We introduce **PEEK**, a system that caches and maintains this orientation knowledge as a context map: a small, constant-sized artifact in the agent's prompt that gives it a persistent peek into the external context. The map is maintained by a programmable cache policy with three modules: a Distiller that extracts transferable knowledge from inference-time signals, a Cartographer that translates it into structured edits, and a priority-based Evictor that enforces a fixed token budget. On long-context reasoning and information aggregation, PEEK improves over strong baselines by 6.3–34.0\% while using 93–145 fewer iterations and incurring 1.7–5.8$\times$ lower cost than the state-of-the-art prompt-learning framework, ACE. On context learning, PEEK improves solving rate and rubric accuracy by 6.0–14.0\% and 7.8–12.1\%, respectively, at 1.4$\times$ lower cost than ACE. These gains generalize across LMs and agent architectures, including OpenAI Codex, a production-grade coding agent. Together, these results show that a context map helps long-context LLM agents interact with recurring external contexts more accurately and efficiently.

Shapley value-based explanations are widely used to attribute model predictions, but standard variants capture associative rather than causal relationships. Recent causally informed Shapley methods add a path-wise decomposition, but suffer from two key limitations: they assign zero attribution to chain-mediated paths, and their per-path values do not align with classical mediation estimands. We propose PE-SHAP, a Shapley-inspired path-wise decomposition built on the portion-eliminated effect. Per-mediator values sum to the total mediated effect at every sample, aggregate to the Natural Indirect Effect at the population level, and match the Path-Specific Effect under additive separability. These are properties prior Shapley-based path-wise methods do not provide. We validate PE-SHAP analytically and through Monte-Carlo experiments on synthetic SCMs with known path effects, and demonstrate it on real-world fairness case studies using the German Credit and COMPAS datasets.

Many applications require joint prediction of interdependent behavioral choices, yet existing models often treat each choice independently (e.g., through parallel prediction heads), overlooking the influence of one on the other. In this work, we propose Progressive Dual-Head Transformer (PDHFormer), a novel framework that performs two-step prediction: the model first estimates one choice and then conditions the second on this upstream estimate through an explicit head-to-head pathway. A shared encoder captures the common structure of two prediction tasks, while the dual-head module explicitly reflects cross-choice dependence. A gated residual mechanism integrated into the embedding layer and the dual-head module further improves the training stability and the prediction performance. Extensive experiments on real-world urban mobility, manufacturing, and food delivery application domains demonstrate that PDHFormer consistently outperforms state-of-the-art machine learning models, deep tabular models, as well as parallel-head Transformer variants across multiple metrics. Moreover, our ablation study confirms that both the proposed progressive dual-head and gated residual mechanism are key contributors to the observed gains in different prediction tasks.

Pose and motion priors play a crucial role in humanoid robotics. Although such priors have been widely studied in human motion recovery (HMR) domain with a range of models, their adoption for humanoid robots remains limited, largely due to the scarcity of high-quality humanoid motion data. In this work, we introduce Pose Distance Fields for Humanoid Robots (PDF-HR), a lightweight prior that represents the robot pose distribution as a continuous and differentiable manifold. Given an arbitrary pose, PDF-HR predicts its distance to a large corpus of retargeted robot poses, yielding a smooth measure of pose plausibility that is well suited for optimization and control. PDF-HR can be integrated as a reward shaping term, a regularizer, or a standalone plausibility scorer across diverse pipelines. We evaluate PDF-HR on various humanoid tasks, including single-trajectory motion tracking, general motion tracking, style-based motion mimicry, and general motion retargeting. Experiments show that this plug-and-play prior consistently and substantially strengthens strong baselines.

The success of vision transformers—especially for generative modeling—is limited by the quadratic cost and weak spatial inductive bias of self-attention. We propose PDE-SSM, a spatial state-space block that replaces attention with a learnable convection–diffusion–reaction partial differential equation. This operator encodes a strong spatial prior by modeling information flow via physically grounded dynamics rather than all-to-all token interactions. Solving the PDE in the Fourier domain yields global coupling with near-linear complexity, delivering a principled and scalable alternative to attention. We integrate PDE-SSM into a flow-matching generative model to obtain the PDE-based Diffusion Transformer PDE-SSM-DiT. Empirically, PDE-SSM-DiT matches or exceeds the performance of state-of-the-art Diffusion Transformers while substantially reducing compute. Our results show that, analogous to 1D settings where SSMs supplant attention, multi-dimensional PDE operators provide an efficient, inductive-bias-rich foundation for next-generation vision models.

Motivated by the success of large language models (LLMs) with broad generalizability and robustness to noisy or unreliable pre-training data, we seek to bring similar capabilities to PDE solvers. In addition, inspired by the posterior predictive mean-based inference mechanism of the in-context learning in prior-data fitted networks (PFNs), we propose PDE-PFN, a prior-data fitted neural solver that approximates the posterior predictive mean of PDE solutions via in-context learning. PDE-PFN builds on a PFN architecture with self- and cross-attention mechanisms of Transformer and is pre-trained on noisy approximate solutions generated by physics-informed neural networks, serving as diverse but not necessarily exact priors. Through experiments on a range of two-dimensional PDEs, we demonstrate that PDE-PFN achieves empirical generalization across heterogeneous equations, robustness under noisy priors, and zero-gradient-update in-context inference capability. Our approach not only outperforms task-specific baselines but also provides a flexible and robust framework for advancing SciML.

Photonic crystal fibres (PCFs) underpin critical applications in telecommunications, sensing, and high-power laser delivery, yet their design remains a manual, expert-intensive process that couples geometric intuition with deep waveguide physics. Vision-language models (VLMs) offer a promising path toward automating this pipeline, but can they truly reason about photonic physics from microscopy images, or do they merely recognise visual patterns? We introduce PCFBench, the first comprehensive benchmark for evaluating VLMs on the complete photonic inverse-design workflow. PCFBench spans 26 tasks organised into six categories of increasing cognitive demand (Geometry Perception, Physics Understanding, Text Generation, Multi-modal Reasoning, Inverse Design, and Code Generation), built from over 220K FDTD simulation samples across 18 structurally diverse fibre families. We evaluate both individual VLMs on isolated tasks and multi-agent systems (MAS) on the full sequential pipeline from perception to design. Our results reveal a striking perception-design gap: current models can identify fibre geometry with moderate success, yet consistently fail to translate that visual understanding into physics-aware design actions. Moreover, cascading agents through the pipeline exposes significant error propagation, where upstream perception mistakes compound into downstream design failures. These findings suggest that neither model scaling nor naive agent chaining suffices; closing the loop demands explicit physical grounding. PCFBench provides the community with a rigorous, multi-dimensional testbed to measure progress toward AI-assisted photonic engineering.

Large Language Models (LLMs) have demonstrated remarkable capabilities across various domains, including software development, education, and technical assistance. However, a fundamental question remains unresolved: can LLMs reason about physical constraints—the rules governing how abstract computations must map onto real-world hardware? This capability is distinct from, and arguably more demanding than, software generation alone, as it requires grounding language understanding in spatial, electrical, and topological constraints. To address this gap, we introduce PCEval (Physical Computing Evaluation), a benchmark and evaluation protocol for physical computing education that enables fully automatic, execution-based assessment of LLMs across both logical and physical implementation artifacts, without human grading. Our evaluation framework assesses LLMs in generating circuits and producing compatible code across varying levels of project complexity. Through comprehensive testing of 14 leading models, PCEval provides the first reproducible and automatically validated empirical assessment of LLMs' ability to reason about fundamental hardware implementation constraints within a simulation environment. Our findings reveal that while LLMs perform well in code generation and logical circuit design, they struggle significantly with physical breadboard layout creation, particularly in managing proper pin connections and avoiding circuit errors. PCEval advances our understanding of AI assistance in hardware-dependent computing environments and establishes a foundation for developing more effective tools to support physical computing education.

Infrared-visible image fusion (IVIF) aims to combine visible and thermal infrared observations for human perception and downstream vision tasks. Visible and thermal infrared observations are physically complementary but unreliable in different ways. Existing deep IVIF methods mainly improve feature interaction and reconstruction, but often lack an explicit local rule for deciding when visible appearance should dominate and when thermal responses should contribute. To address this issue, we reformulate IVIF as local Bayesian posterior rendering, where fusion is controlled by reliability posteriors rather than direct modality mixing. In this paper, we propose Proposal-Context-Detail Fusion (PCDFusion), a visible-anchored framework that implements this formulation through three rendering stages. PCDFusion preserves potentially useful thermal responses, updates exposure-degraded visible regions, and renders reliable band-limited infrared structures into the final luminance. Extensive experiments on public IVIF benchmarks show that PCDFusion improves fusion quality, target recovery, and downstream detection and segmentation performance under challenging illumination. To support reproducibility, the code will be publicly released upon acceptance.