Neural Solvers

Neural network verification has become a key tool for providing formal guarantees on the behaviour of neural networks. However, many verification problems remain computationally intractable in the worst case: even for common adversarial robustness specifications, verification is NP-complete. Here, we explore the application of solver-level warmstarting for neural network verification to exploit information from previous solutions. We study the effect on running time as several properties are modified, including perturbation radii, input data and the networks themselves, using a pipeline that is generalisable and potentially adaptable to state-of-the-art verifiers. Our results show that warmstarting can significantly reduce verification time in most cases. Moreover, warmstarting enables the successful verification of instances that could not be solved from scratch within the given time limit.

Amortizing physics-informed neural networks (PINNs) across related PDEs requires describing each equation to a reusable solver. Coefficient vectors encode numerical parameters in predefined slots, leaving operator and cross-field assignments implicit. We make these relationships explicit in an operator graph, with nodes for fields, derivatives, terms, and residuals and coefficients retained as term attributes. A graph hypernetwork generates diagonal codes that initialize a meta-trained factorized PINN for each target equation. Meta-training and target-specific adaptation use governing equations and prescribed conditions without solution labels. We compare coefficient-vector, DeepSets-based term-set, and graph conditioning by solution accuracy within a fixed adaptation budget. In scalar convection-diffusion-reaction problems, both term-based descriptors improve high-reaction accuracy, with similar performance. In two-field Fisher-KPP, meta-training sees uncoupled and one-way systems; after 3,000 adaptation steps on unseen two-way coupling, the graph's mean final error is 35.7% below the term set and 67.7% below the coefficient vector. In a fixed-structure capacitively coupled plasma model, the coefficient vector performs best. These results support extending coefficient conditioning with explicit equation relationships for physics-based solver adaptation.

Power Flow (PF), Optimal Power Flow (OPF), and State Estimation (SE) are fundamental problems in power system analysis, but solving them is computationally expensive. Graph Neural Networks (GNNs) have been proposed as fast surrogates, yet existing solvers are trained for a single problem at a time, producing narrow models that must be rebuilt for each new task. We propose a more general approach: a single Heterogeneous Residual Gated Graph Convolutional Network that solves all three problems with one shared backbone. Rather than learning one mapping, the model learns a reusable representation of how the network behaves, from which PF, OPF, and SE can each be estimated. Trained jointly on the three problems across diverse topologies and loading conditions, and evaluated on the IEEE 14-bus and 118-bus systems, the shared model matches the accuracy of task-specific GNN solvers and stays robust on unseen loading levels and topologies. These results show that a single model can capture the basic operation of a power network and serve several analysis tasks at once, a first step toward a foundation model for power systems.

Operator learning for parametric partial differential equations (PDEs) typically builds global models over prescribed domains, requiring cross-condition data or costly physics-constrained training. Here we show that the output Jacobian of a neural solution model trained at one condition defines a reusable response space for cross-condition solution variations. We introduce Linearized Subspace Transfer (LST) to exploit this space and recover target solutions by minimizing the target PDE-system residual over response-space coordinates. Because any single response space has finite coverage, Active Transfer Modeling (ATM) uses post-transfer residuals as coverage indicators to selectively acquire response spaces from additional single-condition models. Across six systems, single-condition response spaces supported cross-condition transfer, with enrichment improving accuracy when added spaces expanded representation capacity. Relative to evaluated physics-informed operator baselines, ATM reduced error and offline construction cost, with orders-of-magnitude accuracy gains in representative cases and millisecond-to-second target adaptation. These results establish neural solvers as reusable local parametric models.

Transient glacier simulations with higher-order ice flow require the repeated solution of a nonlinear problem as the geometry evolves. In the online mode of the Instructed Glacier Model, the velocity field is represented by a neural network whose weights are warm-started from the previous time step and updated with a few optimizer iterations. We show that supplying the network with inexpensive input fields derived from low-order ice-flow balances improves this online solver. Unlike residual-based physics-informed neural networks, which incorporate physics through governing-equation penalties in the loss, our approach leaves the governing energy objective unchanged, adding physical structure through the network inputs. Across three real-world glacier configurations, the enriched solver is markedly more robust to solver settings. On the two alpine cases, it also improves the tuned accuracy--runtime trade-off, reducing surface-velocity errors by factors of two to four at fixed runtime and reaching few-percent relative errors with only $10^4$--$10^5$ trainable parameters, far fewer than comparable raw-input baselines. A 300-year Aletsch simulation then completes in under one minute, and the larger Valais domain in about two minutes, on a single GPU---a budget once reserved for much simpler shallow-ice models. Gains are smaller for the fast marine-terminating glacier, where nonlocal stress coupling favors larger or spectral networks. More broadly, the results suggest that enriching a neural solver's inputs with reduced-order physics can make repeated higher-order solves much cheaper, with no training data and no offline training.

Optimal transport (OT) compares distributions and aligns datasets in machine learning, yet unregularized discrete OT requires a linear program with quadratically many transport variables. We propose HELLO, a hierarchical solver that casts large-scale discrete OT as edge localization and uses dual potentials to guide both coarse-to-fine initialization and within-level refinement. Initialization propagates coarse dual potentials across a recursive subsampling hierarchy to assign candidate edges. Refinement then iteratively inserts the largest dual violators in each row and column until the relative KKT residual meets a prescribed tolerance, while budgeted pruning ensures linear memory complexity. For exact-arithmetic refinement, we prove finite termination at a global optimum under a symbolic lexicographic rule. At the million-point scale, HELLO attains lower transport objectives with order-of-magnitude runtime improvements over strong baselines across feature dimensions from single digits to thousands. It further scales to 1.28 million samples per marginal in 8192 dimensions on a single H100, using 41.6 GiB peak GPU memory while satisfying a full relative KKT residual below $10^{-6}$. Beyond standard discrete OT, the framework supports general pairwise costs and serves as a scalable balanced-OT oracle for semi-discrete OT, Gromov--Wasserstein, unbalanced OT, and OT-based Flow Matching.

Sequential reinforcement learning (RL) solvers and global diffusion model (DM) solvers for neural combinatorial optimization exhibit complementary failure modes under an optimization-regret view. The former enjoys small marginal regret in the early construction stage, but suffers from horizon-wise compounding errors with super-linear regret growth; the latter avoids horizon compounding but incurs linear or sublinear regret w.r.t. the dimension of the remaining unsolved subspace. We propose Hybrid Neural Solver for Combinatorial Optimization (HyCO), a hybrid inference algorithm that constructs a solution prefix with an RL solver and adaptively switches to a conditional DM to complete the remaining decisions. To characterize why such hybridization helps, when to trigger the handover, and how to realize it in practice, we first develop a unified error-scaling theoretical framework and prove that, under explicit error-scaling assumptions, i) the hybrid structure achieves strictly lower expected regret than either backbone alone, and ii) there exists a unique optimal trigger step that minimizes the hybrid regret. We then design a lightweight adaptive trigger that combines policy entropy and RL-DM disagreement to detect trajectory-level signals of the regime shift as a practical proxy, since the optimal trigger step is defined at the expected-regret level and is not directly computable on individual trajectories. Experimental results on diverse benchmarks demonstrate that HyCO achieves consistent improvements over both backbones and support the empirical effectiveness of adaptive triggering.

We study the use of graph neural networks (GNNs) for finding approximate ground states of Ising models. Efficiently finding these ground states is of broad significance because many combinatorial optimization problems can be formulated as an Ising model with the appropriate choice of couplings and fields. Exactly solving these problems is hard but there are many good heuristic methods. A lineage of these heuristics build from mean-field approximations: one approach uses the leading eigenvector of an appropriately defined matrix, another is the min-sum algorithm, also known as the energetic cavity method. Without modification, GNNs perform worse than both of these methods. We consider small modifications to the GNN to incorporate these heuristics and find that this considerably improves performance. While the modified approach is competitive against other deep-learning approaches, we still find that simulated annealing is reliably at least as good as deep learning methods for the same computational cost.

Foundation models are transforming business workflows and boosting productivity, yet they remain largely absent from engineering domains such as power system analysis, where strict physical consistency must be enforced. We present GENCO (GEometric Neural Corrective Optimizer), a unified neural solver for steady-state transmission grid analysis that handles power flow (PF), optimal power flow (OPF), and state estimation (SE) within a single architecture and shared network representation. To support advances in neural power system solvers, we introduce the open-source GridFM Development Framework, which standardizes synthetic data generation and training in a low-code environment. We also release large-scale datasets with millions of PF and OPF scenarios across diverse grid topologies to support reproducible benchmarking. We evaluate GENCO on the PFDelta and OPFData benchmarks against state-of-the-art neural solvers and classical solvers, including Newton-Raphson and IPOPT, as well as on real-world Hydro-Québec SCADA data. For large-scale PF, GENCO recovers the full AC operating state, including voltage magnitudes and reactive power that DC-PF cannot provide, while matching DC-PF-level active power-balance residuals. It achieves up to 30x speedups over Newton-Raphson at only 2x the runtime of DC-PF. For OPF, it achieves up to 85x speedups over IPOPT while improving feasibility, optimality, and runtime over DC-OPF. For SE, GENCO is more robust than classical weighted least squares to noisy measurements and network parameter errors, and always returns a high-quality estimate even when weighted least squares fails to converge. Together, the unified architecture and development framework provide a new approach to large-scale steady-state grid analysis, lowering the barrier to entry for power system engineers and marking a step toward Grid Foundation Models.

While the integration of linear constraints has significantly expanded the reach of Answer Set Programming (ASP), existing hybrid solvers often rely on disparate semantic underpinnings that lack a unified logical foundation. We address this gap by introducing a many-sorted variant of the Bound-founded Logic of Here-and-There (HTb), providing a versatile framework capable of characterizing equilibrium models across a wide spectrum of alternative semantics for extensions of ASP with linear constraints. We apply this framework to the setting of difference constraints, focusing on the semantic characterization of clingo[DL]. Central to our approach is the formalization of foundedness for numeric variables. By investigating how different hybrid systems - such as clingo[DL], clingcon, and flingo - justify constraint atoms, we uncover the semantic roots of their varying behaviors. This investigation results in a single, consistent framework that not only formalizes the foundations of current systems like clingo[DL] but also facilitates the rigorous study of program simplifications and the future integration of diverse semantic principles.

Neural solvers are built to deduce, branch, and revise intermediate states. The Lattice Deduction Transformer (LDT) appears to do exactly that. In clue-rich Sudoku, it does not: one forward pass commits essentially the entire grid (every blank cell on standard 6x6, 94-96% on augmented 9x9), turning the iterative solver into a one-shot predictor wrapped in an exact verifier. All hard-slice failures are decided before search begins, when the first pass confidently deletes a value required by the true solution. We call this first-pass poisoning. Adding learned branching, MRV, backtracking, value exclusion, and shared nogoods (CoLT) does not change which Sudoku instances are solved; it cuts repeated invalid derivations 1,497-fold. At the frozen training budget, constraint-graph attention alone matches full-CoLT accuracy, while positional tables recover only under substantially longer training, indicating an optimization and sample-efficiency advantage rather than an absolute capacity difference. The diagnosis predicts two effective interventions. Digit-permutation augmentation raises 9x9 accuracy from below 1% to 96.5 +/- 0.3 across three training seeds on a symmetry-disjoint split. Test-time union over symmetry-transformed passes raises all three hard-slice checkpoints from 72.8-78.9% to 100% without retraining. On from-scratch graph coloring, one-shot behavior disappears and search changes accuracy. In clue-rich completion, LDT-like systems are one-shot amortized predictors rather than learned search procedures: accuracy is determined by calibration and symmetry, while search primarily removes computational waste.

This paper introduces a self-supervised pretraining framework for graph combinatorial optimization specifically designed to address the nature of routing problems like the Traveling Salesman Problem. By utilizing graph contrastive learning with geometric augmentations (specifically, rotations and axial reflections) the model is forced to learn invariant structural representations and global relative distance distributions. Results demonstrate that this pretraining strategy outperforms non-pretrained models across various problem scales. Notably, the hybrid strategy (combining rotation and reflection) achieved a 6.57% improvement in tour length for TSP1000, proving that geometric pretraining is an important inductive bias for effectively scaling neural solvers to high-dimensional instances.

Multi-task neural solvers aim to handle multiple Vehicle Routing Problem (VRP) variants within a unified model, avoiding separate training for each constraint combination. However, VRP variants differ in optimization difficulty, while existing methods lack stage-wise feedback on their training status, making the model biased to some specific variants. Although meta-learning can support adaptive training, it typically requires bi-level optimization and additional gradient updates, increasing computational cost. To address this limitation, we propose LLM-as-Trainer (LaT), a plug-and-play training paradigm that uses a pretrained large language model as an external trainer. LaT periodically analyzes cross-task validation metrics to generate a stage-wise guidance vector. This vector is combined with the current task's constraint vector and injected into each encoder layer, providing the neural solver with additional training information during subsequent policy optimization. Experiments on 16 VRP variants show that LaT improves the solution quality of several state-of-the-art multi-task neural solvers on both trained and unseen variants, supporting the effectiveness and generality of the proposed training paradigm.

Oscillatory Neural Networks (ONNs) present an attractive physics-based computing paradigm rooted in the dynamics of a network of typically fully coupled oscillators aiming to minimize an underlying energy function. In this paper, we propose an ONN-based solver for one well-known constrained combinatorial optimization problem, namely a Sudoku, by formulating the problem as a Graph Coloring problem. By modifying the already existing Graph Coloring solver to a computationally cheaper version and introducing an additional term ensuring the fulfillment of the Sudoku constraints, our solver is shown to significantly outperform the existing HNN- and ONN solvers in terms of accuracy. In particular, we are able to achieve nearly flawless accuracies on $4 \times 4$ as well as rather high accuracies on $9 \times 9$ Sudoku puzzles for different numbers of unknown digits.

Neural autoregressive solvers for the Multi-Attribute Vehicle Routing Problem (MAVRP) reach competitive cost but offer no per-step justification, a problem when dispatchers must validate, accept, or compare them. We open two complementary black boxes in one protocol. On the encoder side, linear probes, spontaneous-organization metrics, rank-based richness measures, and discovered-direction analyses with intervention validation characterize how the latent represents constraint families at the graph, node, and edge level. On the decoder side, three attribution methods (gradient, integrated gradients, DeepLIFT) feed three reading angles: abductive, contrastive against the best feasible alternative, and counterfactual (smallest input change that switches the action or restores feasibility). Explanations are scored on fidelity, concentration, stability, sanity, and actionability. Across six variants combining three encoders (Attention baseline, Unimp, UnimpMoe) with two decoders (Hard-Mask, Recourse), we find that graph inductive bias improves both representational predictability and decoder sanity, that the Mixture-of-Experts encoder represents constraints in a distributed rather than axis-aligned way, and that the Recourse training regime, not merely its softer mask, produces policies that represent infeasibility usefully, exposing make-feasible counterfactuals that Hard-Mask policies fail to produce even when fed infeasible alternatives externally.

Machine learning has shown speedups for numerical methods in physics applications, but integrating Python-based libraries into high-performance C++ solvers creates performance bottlenecks. We present CodeJeNN, which bridges this gap by auto-generating self-contained C++ code from trained Keras models for inference. This eliminates external dependencies through minimal inlined functions, allowing seamless integration into existing frameworks. We describe the Keras-to-C++ workflow, supported architectures, and limitations. CodeJeNN is demonstrated through inference benchmarks against Keras in eager and JIT modes and a CFD test case modeling viscosity in a hydrogen-air mixing layer, showing speedups without sacrificing accuracy.

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/.

High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging. Existing deep learning solvers often rely on repeated automatic differentiation to evaluate differential operators, which can cause instability and amplify derivative errors in high dimensions, while probabilistic methods based on stochastic representations require explicit knowledge of the data-generating dynamics and therefore do not apply to black-box environments. We introduce two types of simulators as data-generating mechanisms, and take a ``representing-then-learning" approach that learns the solutions and their derivatives under settings where the underlying PDE operators are accessible only through simulations and pointwise evaluations. Our representation of derivatives relies on the zeroth-order derivative (ZOD) estimators derived from perturbed Monte Carlo trajectories. This fully model-free approach generates targets for the gradient and Hessian networks using only function evaluations. We provide a statistical learning analysis of the proposed approach, including a bias--variance tradeoff for ZODs. Assuming a standard contraction property of the underlying operator, we establish a non-asymptotic error bound that decomposes the total error into discretization error, approximation error, statistical error, and ZOD bias. Crucially, we derive the sample complexity of the learned representations in (weighted) Sobolev space, characterizing the error up to second-order derivatives. Numerical experiments illustrate the competitive performance of the method in moderate and high dimensions.

The traveling salesman problem (TSP) is a canonical NP-hard combinatorial optimization benchmark that tests the representational capacity and generalization of neural solvers. While non-autoregressive (NAR) approaches offer parallel inference, they often lack sufficient geometric inductive bias and stable training signals, leading to degraded performance under cross-scale and cross-distribution shifts. We propose GeoRouteNet, a geometry-enhanced NAR neural solver for Euclidean TSP. On the model side, GeoRouteNet incorporates centered node features, learnable radial distance basis functions, distance-aware graph attention with explicit edge messaging, LayerNorm-SwiGLU feed-forward blocks, and cross-layer attentive residual mixing. On the training side, we design multi-candidate self-comparison reinforcement learning (MCS-RL), which samples multiple candidate tours per instance, constructs adaptive baselines from greedy and peer candidates, and adds winner-candidate guidance with annealed entropy regularization. On 10,000 random TSP50 instances, GeoRouteNet achieves a 0.32% optimality gap under Beam-1000 decoding. On TSP100, the gap is 1.26%. On 27 stratified TSPLIB EUC_2D instances, the overall gap drops from 17.12% (NAR4TSP reproduction) to 3.60%, while batch inference throughput substantially exceeds that of Concorde and LKH3. Ablation studies confirm that geometric structure enhancement and multi-candidate training are complementary: structure improvements dominate cross-distribution gains, while MCS-RL further stabilizes solution quality when paired with a strong geometric encoder.

Neural combinatorial optimization (NCO) has shown that policies trained by reinforcement can construct strong solutions to NP-hard problems directly from raw instances. What such a policy actually learns, as opposed to what its decoder expresses, remains much less clear. We study this distinction on the vertex-guard Art Gallery Problem, the NP-hard task of choosing polygon vertices from which to observe an entire region. A pointer-network policy is trained from a coverage-aware reward over its own rollouts under the constraint we call geo-free inference: at test time it sees only vertex coordinates, with no visibility computation and no geometric oracle. The policy places guards economically but leaves a tail of under-covered polygons that widens far beyond the training range. To locate the cause, we freeze the trained encoder and read its embeddings with a small single-shot classifier, still geo-free at inference. The classifier closes most of the feasibility gap, in and out of distribution and at up to roughly five times the training range, cutting under-covered polygons by about an order of magnitude at an explicitly reported cost in guard count. We read this as evidence that the reinforcement-trained representation already encodes the geometry required for feasibility, and that residual failures reflect decoder calibration rather than missing knowledge. Probing a frozen encoder thus offers a practical way to ask what a neural combinatorial solver has internalized.

Neural combinatorial optimization has recently achieved strong results on the Euclidean Traveling Salesman Problem (TSP) using generative models such as diffusion and consistency models. State-ofthe-art approaches like FT2T combine fast consistency-based prediction with gradient-based inference time refinement. However, gradient search often incurs significant computational overhead and may not align with the discrete structure of feasible solutions. We introduce Projected Consistency Inference (PCI), a plug-and-play, retraining-free alternative that replaces gradient refinement with structure-aware projections: PCI decodes valid Hamiltonian tours from the consistency model output and applies a lightweight local search (e.g., 2-opt). PCI achieves an average optimality gap (OG) of 0.17% on TSP with 500 cities, and 0.31% on TSP with 1000 cities, outperforming FT2T best settings (OG 0.22% and 0.36%, respectively) while reducing the inference time up to 30 to 40%. PCI also exhibits lower variance and memory usage, and can surpass classical heuristics such as LKH3 in rapid solution generation. Our results demonstrate that structure-aware inference time operations provide a practical and principled path for neural TSP solvers, complementing training time objectives.

Finding optimal solution paths for combinatorial puzzles like the Rubik's Cube, sliding tile puzzles, and Lights Out remains a classical challenge in artificial intelligence. Heuristic search algorithms, such as A* , guarantee path optimality only when using an admissible heuristic-one that never overestimates the true remaining cost-to-go. Deep reinforcement learning (RL) methods like DeepCubeA train deep neural networks to approximate cost-to-go heuristics. However, standard mean-squared error (MSE) training regularly yields overestimations, violating admissibility and compromising solution optimality. In this paper, we introduce a generalizable framework for learning validation-calibrated admissible neural heuristics. We train a value network using an underestimating Admissible Bellman Operator combined with an Asymmetric Loss function to penalize overestimation. To account for residual neural function approximation errors, we propose a post-hoc calibration safety offset computed over validation scrambles. We demonstrate that our calibrated neural heuristics achieve no observed admissibility violations under the evaluation protocol and preserve path optimality in practice while reducing search node expansions by up to 83.0% on a 2 by 2 Rubik's Cube, 19.9% on a 3 by 3 Lights Out grid, and 1.9% on an 8-Puzzle compared to standard analytical baselines.

We propose a five-step diagnostic protocol for residual-trained neural HJB-PIDE solvers with control-dependent Lévy jumps, targeting a general failure mode of neural PDE methods: a learned solution can match headline scalar diagnostics while miscomputing an operator inside its training loss. The protocol pairs each neural solve with at least one from-scratch independent reference, decomposes the Hamiltonian into drift, diffusion, compensator, and nonlocal-integral components across a u-grid, and compares the value function and its low-order derivatives over a (t,x) grid before any argmax comparison. Applied to a standard CRRA-Merton-Variance-Gamma benchmark, it isolates a missing 1/2-mixture factor in the neural method's importance-proposal density that scaled the nonlocal integral by exactly half - a textbook signature of a constant proposal scale error, invisible to longer training, grid refinement, and truncation sweeps. With the bug corrected, four references - two finite-difference solvers with disjoint discretizations, the neural solver, and a semi-analytic scalar baseline obtained from CRRA homogeneity - agree on the optimal control to within ~2%. The constant-coefficient CRRA benchmark collapses by homogeneity to a scalar maximization, so the scalar baseline is the efficient method here; the contribution is the protocol, applicable in principle to non-homogeneous and higher-dimensional settings where neural HJB-PIDE solvers are genuinely needed. The episode is a concrete instance of a broader neural-PDE verification failure: pointwise agreement of a learned value or control can coexist with a systematically wrong nonlocal operator, so per-component and surface-level checks are needed before trusting the argmax policy.

This paper presents the Tensor Product Network (TPNet), a novel neural architecture for efficient and accurate function approximation and PDE solving. The core of the proposal involves constructing the solution explicitly as a linear combination of basis functions integrated into the network, with coefficients determined by a direct least-squares solve, thereby bypassing traditional gradient-based training. The key methodological contribution include: (1) an efficient tensor-product scheme that generates multi-dimensional basis functions from combinations of two sets of subnetwork outputs, significantly reducing model complexity and parameter count while maintaining expressivity; (2) a block time-marching strategy to improve computational efficiency in long-time simulations; and (3) a linear reformulation strategy for handling nonlinear PDEs by treating known nonlinear terms as sources. TPNet achieves superior accuracy and shorter training times than conventional neural network solvers. This performance gain stems from its structured design and deterministic least-squares fitting, which contrast with the iterative, often computationally intensive optimization required by mainstream methods like Physics-Informed Neural Networks (PINNs).

Diffusion-based neural solvers for combinatorial optimization repeatedly re-evaluate dense edge/factor interactions, making inference expensive in wall-clock time and often memory-bound at scale. Inspired by the computational methodologies of many-body physics, we introduce LoRe, a training-free, inference-time drop-in wrapper that enforces per-step interaction-evaluation budgeting: at each iteration, it evaluates only a fixed fraction of interactions by dynamically routing computation to high-conflict or high-uncertainty interactions, instead of using a fixed sparsification (e.g., static kNN graphs or static masks). Under fully inclusive end-to-end wall-clock accounting, LoRe substantially improves scalability on the Maximum Independent Set (MIS) problem, extending feasible inference more than $3\times$ beyond the baseline's out-of-memory limit, delivering a $\sim 8\times$ speedup and a $\sim 12\times$ peak-memory reduction, with solution quality preserved in this regime. Demonstrating cross-task generality on the large-scale Traveling Salesperson Problem (TSP) and zero-shot robustness to topology shifts, LoRe achieves a $\sim 15\times$ speedup at $n=1000$ with a $44\times$ memory reduction and competitive tour quality.

Generalist neural routing solvers have shown great potential in solving diverse vehicle routing problems (VRPs) with a unified model. However, existing solvers are typically limited to symmetric settings or degrade in performance when switching to asymmetric settings due to input inconsistencies or inherent structural differences, substantially limiting their practicality in real-world scenarios that encompass both scenarios. To address this limitation, we define the spatial position of each node based on the relative distances to a specific set of pivots and further propose a Spatial Pivot-Aligned Coordinate-free Embedding (SPACE) framework that unifies node representation and solution generation across symmetric and asymmetric VRPs. Specifically, we construct a bidirectional Frechet representation using a novel furthest pivot sampling strategy to enable invariant node representations across distinct problem settings. Furthermore, we introduce a weight-decomposed adaptive decoding mechanism that decouples geometric perception from problem representations, mitigating the overfitting of constraint decisions to a specific geometry setting. Extensive experiments on 110 VRP variants, comprising 55 symmetric problems and their asymmetric counterparts, demonstrate that SPACE achieves promising zero-shot generalization in both symmetric and asymmetric VRPs.

Existing neural solvers for Multi-Objective Combinatorial Optimization Problems (MOCOPs) commonly adopt decomposition-based strategies that scalarize an MOCOP into multiple subproblems associated with distinct weight vectors. However, they either inject weights only once during decoding, limiting weight-conditioned context modeling, or primarily during encoding, causing weight-signal dilution during decoding. Moreover, preference optimization methods rely on purely random sampling to construct solution pairs for training solvers, which often produces less informative pairs and thus leads to low training effectiveness. To better address these limitations, we propose an efficient Weight-Conditioned neural solver (WeCon). Specifically, we design an encoder layer with three attention blocks and our proposed Gated Residual Fusion (GRF) block to facilitate harmonious interaction between instance features and weights, thereby generating informative weight-conditioned context. We further introduce a plug-and-play Residual Fusion (RF) block in the decoder to alleviate weight-signal dilution. Finally, we propose Efficient Preference Optimization (EPO), which constructs high-quality solutions, thereby generating more informative pairs to improve training effectiveness. Experiments on four MOCOP variants across different problem scales and distribution patterns demonstrate that WeCon achieves HyperVolume (HV) values comparable to SOTA solver POCCO-W, while reducing inference time by 40%. Ablation studies validate the contributions of all designs.

Although Vehicle Routing Problems (VRP) are essential to many real-world systems, they remain computationally intractable at scale due to their combinatorial complexity. Traditional heuristics rely on handcrafted rules for local improvements and occasional \textit{jumps} to escape local minima, but often struggle to generalize across diverse instances. We introduce \textbf{COAgents}, a cooperative multi-agent framework that models the search process as a graph: nodes represent solutions, and edges correspond to either local refinements or large perturbations for diversification (i.e., jumps). A \textit{Partial Search Graph} (PSG) is dynamically constructed during search, enabling COAgents to train a Node Selection Agent and a Move Selection Agent to guide intensification, and a Jump Agent to trigger well-timed explorations of new regions. Unlike end-to-end learning approaches, COAgents cleanly separates problem-agnostic search control from compact domain-specific encoding, facilitating adaptability across tasks. Extensive experiments on the CVRP and VRPTW benchmarks show that COAgents remains competitive with several learn-to-search baselines on CVRP and sets a new state of the art among learning-based methods on the more challenging VRPTW instances, reducing the gap to the best-known solutions by 14% at $N\!=\!100$ and 44% at $N\!=\!50$ relative to the strongest neural solver (POMO), and by 21% and 40% respectively relative to ALNS. Code is available at https://github.com/mahdims/COAgents.

The multi-path Traveling Salesman Problem with stochastic travel costs arises in hybrid vehicle routing applications designed for Smart City and City Logistics, where multiple paths exist between each pair of locations. Travel times along these paths are typically affected by real-time traffic conditions and therefore modeled as stochastic. The objective of the problem is to determine a Hamiltonian tour that minimizes the expected total travel cost under uncertainty. In this work, we adopt a two-stage stochastic programming formulation. In the first stage, a predefined route specifying the sequence of locations to be visited is determined, while taking into consideration a second-stage recourse problem that selects the optimal path from the feasible set of alternative paths for each pair of locations, once real-time traffic conditions are realized. To reduce the computational burden imposed by the large number of scenarios required to capture travel time uncertainty, the innovation of this work is the integration of neural network-based surrogate models to approximate the expected value of the second-stage recourse problem. Different architectures and training strategies for the neural networks are proposed and analyzed, with performance evaluated in terms of computation time, solution quality, and generalization capability. Preliminary findings demonstrate the enhanced scalability and practical applicability of the approach for complex vehicle routing problems under uncertainty.

A common critique of neural combinatorial-optimization solvers is that they are less energy-efficient than CPU metaheuristics, given the operational energy cost of training them on GPUs. This paper examines the inferential step from "training is expensive" to "neural solvers are net-inefficient", which is where the critique actually goes wrong. Training the network costs a large fixed amount of GPU energy; running the metaheuristic costs a small amount of CPU energy on every instance, repeated as long as the solver is deployed. The two are not commensurable until a deployment volume is fixed. We define the Amortized Efficiency Threshold (AET) as the deployment volume above which a neural solver breaks even with a heuristic baseline in total energy or carbon, under an explicit constraint on solution quality. We show that the cumulative-energy ratio between the two solvers tends to a constant strictly below one whenever the network wins per instance, and that this limit does not depend on how the training cost was measured. An embodied-carbon term amortizes hardware fabrication symmetrically on both sides. We instantiate the framework on the CVRP environment at n=50 customers with the attention-based autoregressive solver of Kool et al. (2019), trained for 100 epochs on 20,000 instances over five random seeds, and HGS via PyVRP as the heuristic baseline. The measured operational crossover sits near 4.56e3 deployed instances at the median of a six-point baseline-budget sweep; the per-instance neural-to-heuristic ratio is 2.29e-3. The contribution is the framework, the open instrumentation, and the end-to-end measurement protocol. Code and benchmark pipeline are available at https://github.com/sohaibafifi/aet.

Implicit solvent models are widely used to decrease the number of solvent degrees of freedom and enable the calculation of solvation energetics without water molecules. However, its accuracy often falls short compared to explicit models. Recent advancements in neural potentials have shown promise in drug discovery, but transferability remains a persistent challenge. Here, we introduce the Protein Hydration Neural Network (PHNN), an implicit solvent model that extends analytical continuum solvation by learning transferable corrections to model parameters instead of applying post hoc adjustments to final energies. The model is explicitly designed to maximize data efficiency by leveraging physical priors embedded in the data. We demonstrate that PHNN improves accuracy relative to traditional analytical methods and maintains predictive accuracy on out-of-domain protein systems.

Heavy-Encoder-Light-Decoder (HELD) neural routing solvers have emerged as a promising paradigm due to their broad applicability across multiple vehicle routing problems (VRPs). However, they typically struggle with VRP variants with complex constraints. To address this limitation, this paper systematically revisits existing neural solvers from the perspective of the generation mechanism for state embeddings (i.e., query vector prior to compatibility calculation) during decoding. We identify that current mechanisms restrict the observation space during attention computation, introducing a key bottleneck to achieving high-quality solutions. Through detailed empirical analysis, we demonstrate the necessity of preserving a global observation space. To overcome the constraint-agnostic drawback inherent to global observation spaces, we propose a simple yet powerful Constraint-Aware Residual Modulation (CARM) module. By adaptively modulating the context embedding with constraint-relevant variables, CARM effectively enhances constraint awareness, enabling the neural solver to fully leverage the global observation space and generate an efficient state embedding. Extensive experimental results across two single-task and five multi-task neural routing solvers confirm that the CARM module consistently boosts baseline performance. Notably, solvers equipped with our CARM achieve substantial improvements in scaling to large-scale instances and in generalizing to unseen VRP variants. These findings provide valuable insights for the architectural design of neural routing solvers.

Crash simulation is a cornerstone of modern vehicle development because it reduces the need for costly physical prototypes, accelerates safety-driven design iteration, and increasingly supports virtual testing workflows. At the same time, modeling structural crash mechanics remains exceptionally challenging: the response is governed by nonlinear contact, large deformation, material plasticity, failure, and complex multi-body interactions evolving over space and time on high-resolution finite-element meshes. In this work, we introduce CarCrashNet, a public high-fidelity open-source benchmark for data-driven structural crash simulation. CarCrashNet combines component-scale and full-vehicle simulations in a multi-modal format, including more than 14,000 bumper-beam pole-impact simulations with varying geometry, materials, and boundary conditions, together with 825 full-vehicle crash simulations built from three industry-standard vehicle models of increasing structural complexity: Dodge Neon, Toyota Yaris, and Chevrolet Silverado. To establish the reliability of the benchmark, we validate our open-source finite-element workflow based on OpenRadioss against both experimental crash data and the commercial solver Ansys LS-DYNA. We also introduce CrashSolver, a machine-learning model designed for full-vehicle crash prediction from high-resolution finite-element crash data. We further perform extensive benchmarking across the released datasets and evaluate CrashSolver against state-of-the-art geometric deep learning and transformer-based neural solvers. Our results position CarCrashNet as a foundation for reproducible research in structural simulation, crashworthiness modeling, and AI-driven virtual crash testing. The dataset is available at https://github.com/Mohamedelrefaie/CarCrashNet.

In this paper, we introduce INEUS, a meshfree iterative neural solver for partial integro-differential equations (PIDEs). The method replaces the explicit evaluation of nonlocal jump integrals with single-jump sampling and reformulates PIDE solving as a sequence of recursive regression problems. Like Physics-Informed Neural Networks (PINNs), INEUS learns global solutions over the entire space-time domain, yet it offers a more efficient treatment of nonlocal terms and avoids the computationally expensive differentiation of full PIDE residuals. These features make INEUS particularly well suited for high-dimensional PDEs and PIDEs. Supported by a contraction-based convergence proof for linear PIDEs, our numerical experiments show that INEUS delivers accurate and scalable solutions for various high-dimensional linear and nonlinear examples.

Rapid aerodynamic evaluation is crucial for modern vehicle design, yet existing neural operators struggle to capture intricate spatial correlations. We propose the rotary-enhanced transformer operator (RETO), a novel neural solver featuring a dual-stage spatial awareness mechanism: sinusoidal-cosine encodings for global referencing and rotary positional encodings (RoPE) for relative displacements. RoPE encodes spatial relations via unitary rotations, enforcing translation invariance and enhancing local gradient resolution. RETO is validated on ShapeNet and the high-fidelity DrivAerML benchmark. On ShapeNet, RETO achieves a relative $L_2$ error of 0.063, outperforming RegDGCNN at 0.125 and representing a 16% improvement over the Transolver baseline, which yields an error of 0.075. These performance gains are further amplified on the DrivAerML dataset, where RETO achieves relative $L_2$ errors of 0.089 for surface pressure and 0.097 for velocity. In comparison, Transolver results in errors of 0.116 and 0.121 for the same metrics, indicating that RETO achieves precision enhancements of 23% and 19%, respectively. For comprehensive comparison, the surface pressure and velocity errors for AB-UBT are 0.102 and 0.124, while RegDGCNN yields 0.235 and 0.312, respectively. Information-theoretical analysis shows that the entropy peak of RETO at 0.35 is significantly lower than that of Transolver at 0.75 under $10^4$ resolution, indicating a focused attentional mechanism capable of preserving localized gradients against global diffusion.

Standard physics-informed neural networks (PINNs) struggle to simulate highly oscillatory Helmholtz solutions in heterogeneous media because pointwise minimization of second-order PDE residuals is computationally expensive, biased toward smooth solutions, and requires artificial absorbing boundary layers to restrict the solution. To overcome these challenges, we introduce a Green-Integral (GI) neural solver for the acoustic Helmholtz equation. It departs from the PDE-residual-based formulation by enforcing wave physics through an integral representation that imposes a nonlocal constraint. Oscillatory behavior and outgoing radiation are encoded directly through the integral kernel, eliminating second-order spatial derivatives and enforcing physical solutions without additional boundary layers. Theoretically, optimizing this GI loss via a neural network acts as a spectrally tuned preconditioned iteration, enabling convergence in heterogeneous media where the classical Born series diverges. By exploiting FFT-based convolution to accelerate the GI loss evaluation, our approach substantially reduces GPU memory usage and training time. However, this efficiency relies on a fixed regular grid, which can limit local resolution. To improve local accuracy in strong scattering regions, we also propose a hybrid GI+PDE loss, enforcing a lightweight Helmholtz residual at a small number of nonuniformly sampled collocation points. We evaluate our method on seismic benchmark models characterized by structural contrasts and subwavelength heterogeneity at frequencies up to 20Hz. GI-based training consistently outperforms PDE-based PINNs, reducing computational cost by over a factor of ten. In models with localized scattering, the hybrid loss yields the most accurate reconstructions, providing a stable, efficient, and physically grounded alternative.

This paper presents the hybrid solver for a $CO_2$ sequestration problem. The solver uses the IGA-ADS (IsoGeometric Analysis Alternating Directions solver) to compute the saturation scalar field update using the explicit method, and CRVPINN (Collocation-based Robust Variational Physics Informed Neural Networks solver) to compute the pressure scalar field. The study focuses on simulating the physical behavior of $CO_2$ in porous structures, excluding chemical reactions. The mathematical model is based on Darcy's Law. The CRVPINN is pretrained on the initial pressure configuration, and the time step pressure updates require only 100 iterations of the Adam method per time step. We compare our hybrid IGA-ADS solver, coupled with the CRVPINN method, with a baseline of the IGA-ADS solver coupled with the MUMPS direct solver. Our hybrid solver is over 3 times faster on a single computational node from the ARES cluster of ACK CYFRONET. Future work includes extensive testing, inverse problem solving, and potential application to $H_2$ storage problems.

Hyperbolic conservation laws govern a wide range of transport-driven dynamics featuring shocks, contact discontinuities, and complex wave interactions, posing distinct challenges for deep-learning-based surrogate modeling. While classical numerical methods provide robust and physically admissible solutions, their computational cost restricts applicability in many-query tasks such as parametric studies and design optimization. Conversely, existing neural surrogates offer rapid inference but often fail to respect intrinsic PDE structures, leading to non-physical artifacts, rollout instability, and poor generalization. We present an interpretable, structure-preserving graph neural solver that bridges classical numerical principles with graph neural networks (GNNs). The network is designed as a learned reconstruction-and-flux operator rather than a black-box state updater, thereby inherently preserving key properties such as local conservation and upwinding. Inspired by Arbitrary high-order DERivatives schemes, we further recast message-passing GNNs as high-order space-time predictors, enabling conservative and stable neural updates with large time steps. Evaluation is performed on challenging supersonic flow benchmarks spanning broad parametric variations in geometry, initial/boundary conditions, and flow regimes. The neural solver achieves superior long-horizon rollout stability and accuracy compared with strong surrogate baselines, outperforms low-order discretizations, and delivers orders-of-magnitude runtime speedups over high-resolution simulations.

Deep learning has emerged as a transformative tool for the neural surrogate modeling of partial differential equations (PDEs), known as neural PDE solvers. However, scaling these solvers to industrial-scale geometries with over $10^8$ cells remains a fundamental challenge due to the prohibitive memory complexity of processing high-resolution meshes. We present Transolver-3, a new member of the Transolver family as a highly scalable framework designed for high-fidelity physics simulations. To bridge the gap between limited GPU capacity and the resolution requirements of complex engineering tasks, we introduce two key architectural optimizations: faster slice and deslice by exploiting matrix multiplication associative property and geometry slice tiling to partition the computation of physical states. Combined with an amortized training strategy by learning on random subsets of original high-resolution meshes and a physical state caching technique during inference, Transolver-3 enables high-fidelity field prediction on industrial-scale meshes. Extensive experiments demonstrate that Transolver-3 can handle meshes with over 160 million cells, achieving impressive performance across three challenging simulation benchmarks, including aircraft and automotive design tasks. Code is available at https://github.com/thuml/Transolver-3.

Inverse Kinematics (IK) is a core problem in robotics, in which joint configurations are found to achieve a (collision-free) desired end-effector pose. Modern IK solvers face a fundamental trade-off: analytical methods are fast but lack generality, while numerical optimization-based methods are broadly applicable but prone to local minima and high computational costs. To overcome this challenge, we introduce HJCD-IK, a GPU-accelerated, sampling-based hybrid solver. By pairing a novel orientation-aware greedy coordinate descent initialization with Jacobian-based polishing and a parallel collision filter, our method achieves up to order-of-magnitude gains in speed and accuracy over state-of-the-art solvers, consistently finding collision-free solutions on the accuracy-latency Pareto frontier, while producing a diverse distribution of high-quality samples. We validate our solver on a physical Franka manipulator and release our code open-source.

The primary paradigm in Neural Combinatorial Optimization (NCO) consists of construction methods, where a neural network is trained to sequentially add one solution component at a time until a complete solution is formed. We observe that the typical changes to the state between two steps are small, since usually only the node added to the solution is removed from the state. An efficient model should be able to reuse computation from prior steps. To that end, we propose a recurrent encoder that computes state embeddings based not only on the current state but also on embeddings from the previous state. We show that this recurrent encoder can achieve equivalent or better performance than a non-recurrent encoder even with $3\times$ fewer layers, thus significantly improving latency. We demonstrate our findings on three different problems: the Traveling Salesman Problem (TSP), the Capacitated Vehicle Routing Problem (CVRP), and the Orienteering Problem (OP), and integrate the models into a large neighborhood search algorithm to showcase the practical relevance of our findings.

Physics informed neural networks (PINNs) represent a very popular class of neural solvers for partial differential equations. In practice, one often employs stochastic gradient descent type algorithms to train the neural network. Therefore, the convergence guarantee of stochastic gradient descent is of fundamental importance. In this work, we establish the linear convergence of stochastic gradient descent / flow in training over-parameterized two layer PINNs with a general class of activation functions for solving one model second-order elliptic problem, i.e., the Poisson equation, in the sense of high probability. These results extend the existing result [20] in which gradient descent was analyzed. The challenge of the analysis lies in handling the dynamic randomness introduced by stochastic optimization methods. The key of the analysis lies in ensuring the positive definiteness of suitable Gram matrices during the training. The analysis sheds insight into the dynamics of the optimization process, and provides guarantees on physics informed neural networks trained by stochastic algorithms.

We analyze a neural semi-discrete method for high-dimensional first-order Hamilton-Jacobi-Bellman (HJB) equations with known or learned dynamics. Centered differences and an artificial viscosity $Nh=O(h)$ define a monotone operator evaluated through $2d+1$ shifted network queries; policy iteration solves the resulting Bellman equation without a tensor grid. At fixed $h$, the sharp componentwise condition $\max_i|f_i|\le2N$ turns every frozen-policy operator into a nearest-neighbor Markov-chain generator with a policy-independent total jump rate. Uniformization gives whole-space well-posedness for measurable feedbacks, an explicit Poisson-tail bound on the numerical domain of dependence, and boundary-free localization. The representation also yields a posteriori policy-evaluation bounds that account for residual and learned-model errors. A greedy-gap analysis controls inexact policy iteration at fixed $h$; a separate consistency estimate connects the semi-discrete equation to the continuous HJB equation. Experiments reproduce the extremal tail, show rates consistent with $O(\sqrt h)$ and nearly $h$-independent exact-policy-iteration decay, and assess empirical estimator effectivity. A nonsmooth example shows that the continuous residual can miss a non-viscosity solution, whereas the shifted residual detects the defect. Further tests provide a structured interval-verified certificate calibration, an early-budget benefit of policy freezing for bang-bang control, and learned-dynamics diagnostics. A structured nonlinear problem with active compact-control constraints is tested against a manufactured semi-discrete reference through $d=1024$.