arXiv subjects

Zhongkai Hao

Publications and source records attributed to Zhongkai Hao.

At least 19 recordsLinked to original sources

arXiv2026

SE-GoS: Self-Evolving Graph-of-Skills for Skill Library at Scale

Modern LLM agents increasingly rely on reusable skills, yet as skill libraries scale to thousands of entries, effective retrieval becomes a bottleneck. Graph-of-Skills (GoS) addresses this challenge by exploiting dependency-aware graph structure for scalable skill retrieval, while SkillDAG further demonstrates that skill graphs can accumulate execution-backed structure online. However, these approaches leave open whether historical execution traces can be systematically distilled into a better retrieval graph that generalizes to unseen tasks. We present Self-Evolving Graph-of-Skills (SE-GoS), a training-free framework that evolves an existing GoS graph from execution traces while preserving the original retrieval pipeline. SE-GoS performs three complementary updates: topology evolution that discovers and prunes skill relationships from execution evidence, edge-weight evolution that reinforces retrieval-relevant relationships based on historical effectiveness, and description evolution that optimizes retrieval-facing skill descriptions using execution feedback. Across three LLMs on SkillsBench, SE-GoS consistently improves task reward while reducing input tokens relative to full skill loading, with gains varying across model families. In a representative setting, one evolution round improves reward from 52.4\% to 59.4\% while reducing input tokens by approximately one-third relative to full skill loading, and the resulting graph transfers to a disjoint held-out split with a 5.4-point improvement over the static GoS baseline. These results show that skill graphs can be improved from execution experience without model training, changes to the retrieval algorithm, or modifications to skill content, turning a static retrieval graph into an evolving retrieval infrastructure.

cs.AI↗

arXiv2026

AOT-POT: Adaptive Operator Transformation for Large-Scale PDE Pre-training

Pre-training neural operators on diverse partial differential equation (PDE) datasets has emerged as a promising direction for building general-purpose surrogate models in scientific machine learning. However, the inherent complexity and structural diversity of PDE solution operators make multi-PDE pre-training fundamentally challenging. Existing methods mainly address this by increasing model capacity, while leaving the target solution operators unchanged. Inspired by classical numerical analysis, we instead propose to transform complex and diverse solution operators into simpler, better-aligned forms that are easier to model jointly. Since the optimal transformation varies across PDE types, it must be adaptive and input-dependent, allowing a single neural operator to approximate an entire family of operators. We instantiate this idea as AOT-POT (adaptive operator-transformation for pre-training operator transformer), which expands hidden representations into multiple parallel streams, adaptively aggregates and redistributes them before and after each sub-layer, and mixes streams through Sinkhorn-projected doubly stochastic matrices for stable training. These mechanisms together reshape diverse solution operators into a unified form that can be effectively modeled by a single architecture. Empirically, AOT-POT achieves state-of-the-art performance on 12 PDE benchmarks with only 3\% additional parameters, reducing relative L2 error by up to 77.6\% (40.9\% on average). Fine-tuning AOT-POT further reduces L2 error by up to 92\% on in-domain PDEs and 89\% on out-of-domain PDEs (unseen types during pre-training), demonstrating that adaptive operator transformation is an effective and complementary direction for advancing PDE foundation models beyond simply scaling model capacity.

cs.LG↗

arXiv2026

Discovering Physical Directions in Weight Space: Composing Neural PDE Experts

Recent advances in neural operators have made partial differential equation (PDE) surrogate modeling increasingly scalable and transferable through large-scale pretraining and in-context adaptation. However, after a shared operator is fine-tuned to multiple regimes within a continuous physical family, it remains unclear whether the resulting weight-space updates merely form isolated regime experts or reveal reusable physical structure. Starting from a shared family anchor, we fine-tune low- and high-regime endpoint experts and show that their updates can be separated into a family-shared adaptation and a direction aligned with the underlying physical parameter. This separation reinterprets endpoint experts as finite-difference probes of a local physical direction in weight space, explaining why static averaging can interpolate between regimes but attenuates endpoint-specific physics. Building on this perspective, we propose Calibration-Conditioned Merge (CCM), a post-hoc coordinate readout method for composing neural PDE experts along this physical direction. Given physical metadata, a calibrated coordinate mapping, or a short observed rollout prefix, CCM infers the target composition coordinate and deploys a single merged checkpoint for the remaining rollout. We evaluate CCM on the reaction--diffusion system, viscosity-parameterized two-dimensional Navier--Stokes equations, and radial dam-break dynamics. Across these benchmarks, CCM achieves its strongest gains in extrapolative regimes, reducing out-of-distribution rollout error relative to the family anchor by 54.2%, 42.8%, and 13.8%, respectively. Further experiments across FNO scales, a DPOT-style backbone, and ablations confirm that endpoint fine-tuning is not arbitrary checkpoint drift, but reveals a calibratable physical direction for training-free transfer across PDE regimes.

cs.LG↗

arXiv2026

Helix: Evolutionary Reinforcement Learning for Open-Ended Scientific Problem Solving

Large language models (LLMs) with reasoning abilities have demonstrated growing promise for tackling complex scientific problems. Yet such tasks are inherently domain-specific, unbounded and open-ended, demanding exploration across vast and flexible solution spaces. Existing approaches, whether purely learning-based or reliant on carefully designed workflows, often suffer from limited exploration efficiency and poor generalization. To overcome these challenges, we present HELIX -- a Hierarchical Evolutionary reinforcement Learning framework with In-context eXperiences. HELIX introduces two key novelties: (i) a diverse yet high-quality pool of candidate solutions that broadens exploration through in-context learning, and (ii) reinforcement learning for iterative policy refinement that progressively elevates solution quality. This synergy enables the discovery of more advanced solutions. On the circle packing task, HELIX achieves state-of-the-art result with a sum of radii of 2.63598308 using only a 14B model. Across standard machine learning benchmarks, HELIX further surpasses GPT-4o with a carefully engineered pipeline, delivering an average F1 improvement of 5.95 points on the Adult and Bank Marketing datasets.

cs.LG↗

arXiv2026

Pretrain Finite Element Method: A Pretraining and Warm-start Framework for PDEs via Physics-Informed Neural Operators

We propose a Pretrained Finite Element Method (PFEM),a physics driven framework that bridges the efficiency of neural operator learning with the accuracy and robustness of classical finite element methods (FEM). PFEM consists of a physics informed pretraining stage and an optional finetuning stage. In the pretraining stage, a neural operator based on the Transolver architecture is trained solely from governing partial differential equations, without relying on labeled solution data. The model operates directly on unstructured point clouds, jointly encoding geometric information, material properties, and boundary conditions, and produces physically consistent initial solutions with extremely high computational efficiency. PDE constraints are enforced through explicit finite element, based differentiation, avoiding the overhead associated with automatic differentiation. In the fine-tuning stage, the pretrained prediction is used as an initial guess for conventional FEM solvers, preserving their accuracy, convergence guarantees, and extrapolation capability while substantially reducing the number of iterations required to reach a prescribed tolerance. PFEM is validated on a broad range of benchmark problems, including linear elasticity and nonlinear hyperelasticity with complex geometries, heterogeneous materials, and arbitrary boundary conditions. Numerical results demonstrate strong generalization in the pretraining stage with relative errors on the order of 1\%, and speedups of up to one order of magnitude in the fine-tuning stage compared to FEM with zero initial guesses.

math.NA↗

arXiv2025

An Efficient Graph-Transformer Operator for Learning Physical Dynamics with Manifolds Embedding

Accurate and efficient physical simulations are essential in science and engineering, yet traditional numerical solvers face significant challenges in computational cost when handling simulations across dynamic scenarios involving complex geometries, varying boundary/initial conditions, and diverse physical parameters. While deep learning offers promising alternatives, existing methods often struggle with flexibility and generalization, particularly on unstructured meshes, which significantly limits their practical applicability. To address these challenges, we propose PhysGTO, an efficient Graph-Transformer Operator for learning physical dynamics through explicit manifold embeddings in both physical and latent spaces. In the physical space, the proposed Unified Graph Embedding module aligns node-level conditions and constructs sparse yet structure-preserving graph connectivity to process heterogeneous inputs. In the latent space, PhysGTO integrates a lightweight flux-oriented message-passing scheme with projection-inspired attention to capture local and global dependencies, facilitating multilevel interactions among complex physical correlations. This design ensures linear complexity relative to the number of mesh points, reducing both the number of trainable parameters and computational costs in terms of floating-point operations (FLOPs), and thereby allowing efficient inference in real-time applications. We introduce a comprehensive benchmark spanning eleven datasets, covering problems with unstructured meshes, transient flow dynamics, and large-scale 3D geometries. PhysGTO consistently achieves state-of-the-art accuracy while significantly reducing computational costs, demonstrating superior flexibility, scalability, and generalization in a wide range of simulation tasks.

cs.CE↗

arXiv2025

A$^2$Search: Ambiguity-Aware Question Answering with Reinforcement Learning

Recent advances in Large Language Models (LLMs) and Reinforcement Learning (RL) have led to strong performance in open-domain question answering (QA). However, existing models still struggle with questions that admit multiple valid answers. Standard QA benchmarks, which typically assume a single gold answer, overlook this reality and thus produce inappropriate training signals. Existing attempts to handle ambiguity often rely on costly manual annotation, which is difficult to scale to multi-hop datasets such as HotpotQA and MuSiQue. In this paper, we present A$^2$Search, an annotation-free, end-to-end training framework to recognize and handle ambiguity. At its core is an automated pipeline that detects ambiguous questions and gathers alternative answers via trajectory sampling and evidence verification. The model is then optimized with RL using a carefully designed $\mathrm{AnsF1}$ reward, which naturally accommodates multiple answers. Experiments on eight open-domain QA benchmarks demonstrate that A$^2$Search achieves new state-of-the-art performance. With only a single rollout, A$^2$Search-7B yields an average $\mathrm{AnsF1}@1$ score of $48.4\%$ across four multi-hop benchmarks, outperforming all strong baselines, including the substantially larger ReSearch-32B ($46.2\%$). Extensive analyses further show that A$^2$Search resolves ambiguity and generalizes across benchmarks, highlighting that embracing ambiguity is essential for building more reliable QA systems. Our code, data, and model weights can be found at https://github.com/zfj1998/A2Search

cs.CL↗

arXiv2025

AnyPos: Automated Task-Agnostic Actions for Bimanual Manipulation

Learning generalizable manipulation policies hinges on data, yet robot manipulation data is scarce and often entangled with specific embodiments, making both cross-task and cross-platform transfer difficult. We tackle this challenge with task-agnostic embodiment modeling, which learns embodiment dynamics directly from task-agnostic action data and decouples them from high-level policy learning. By focusing on exploring all feasible actions of the embodiment to capture what is physically feasible and consistent, task-agnostic data takes the form of independent image-action pairs with the potential to cover the entire embodiment workspace, unlike task-specific data, which is sequential and tied to concrete tasks. This data-driven perspective bypasses the limitations of traditional dynamics-based modeling and enables scalable reuse of action data across different tasks. Building on this principle, we introduce AnyPos, a unified pipeline that integrates large-scale automated task-agnostic exploration with robust embodiment modeling through inverse dynamics learning. AnyPos generates diverse yet safe trajectories at scale, then learns embodiment representations by decoupling arm and end-effector motions and employing a direction-aware decoder to stabilize predictions under distribution shift, which can be seamlessly coupled with diverse high-level policy models. In comparison to the standard baseline, AnyPos achieves a 51% improvement in test accuracy. On manipulation tasks such as operating a microwave, toasting bread, folding clothes, watering plants, and scrubbing plates, AnyPos raises success rates by 30-40% over strong baselines. These results highlight data-driven embodiment modeling as a practical route to overcoming data scarcity and achieving generalization across tasks and platforms in visuomotor control. Project page: https://embodiedfoundation.github.io/vidar_anypos.

cs.CV↗

arXiv2025

Accelerating PDE-Constrained Optimization by the Derivative of Neural Operators

PDE-Constrained Optimization (PDECO) problems can be accelerated significantly by employing gradient-based methods with surrogate models like neural operators compared to traditional numerical solvers. However, this approach faces two key challenges: (1) **Data inefficiency**: Lack of efficient data sampling and effective training for neural operators, particularly for optimization purpose. (2) **Instability**: High risk of optimization derailment due to inaccurate neural operator predictions and gradients. To address these challenges, we propose a novel framework: (1) **Optimization-oriented training**: we leverage data from full steps of traditional optimization algorithms and employ a specialized training method for neural operators. (2) **Enhanced derivative learning**: We introduce a *Virtual-Fourier* layer to enhance derivative learning within the neural operator, a crucial aspect for gradient-based optimization. (3) **Hybrid optimization**: We implement a hybrid approach that integrates neural operators with numerical solvers, providing robust regularization for the optimization process. Our extensive experimental results demonstrate the effectiveness of our model in accurately learning operators and their derivatives. Furthermore, our hybrid optimization approach exhibits robust convergence.

cs.LG↗

arXiv2025

Exploratory Diffusion Model for Unsupervised Reinforcement Learning

Unsupervised reinforcement learning (URL) aims to pre-train agents by exploring diverse states or skills in reward-free environments, facilitating efficient adaptation to downstream tasks. As the agent cannot access extrinsic rewards during unsupervised exploration, existing methods design intrinsic rewards to model the explored data and encourage further exploration. However, the explored data are always heterogeneous, posing the requirements of powerful representation abilities for both intrinsic reward models and pre-trained policies. In this work, we propose the Exploratory Diffusion Model (ExDM), which leverages the strong expressive ability of diffusion models to fit the explored data, simultaneously boosting exploration and providing an efficient initialization for downstream tasks. Specifically, ExDM can accurately estimate the distribution of collected data in the replay buffer with the diffusion model and introduces the score-based intrinsic reward, encouraging the agent to explore less-visited states. After obtaining the pre-trained policies, ExDM enables rapid adaptation to downstream tasks. In detail, we provide theoretical analyses and practical algorithms for fine-tuning diffusion policies, addressing key challenges such as training instability and computational complexity caused by multi-step sampling. Extensive experiments demonstrate that ExDM outperforms existing SOTA baselines in efficient unsupervised exploration and fast fine-tuning downstream tasks, especially in structurally complicated environments.

cs.LG↗

arXiv2024

PAPM: A Physics-aware Proxy Model for Process Systems

In the context of proxy modeling for process systems, traditional data-driven deep learning approaches frequently encounter significant challenges, such as substantial training costs induced by large amounts of data, and limited generalization capabilities. As a promising alternative, physics-aware models incorporate partial physics knowledge to ameliorate these challenges. Although demonstrating efficacy, they fall short in terms of exploration depth and universality. To address these shortcomings, we introduce a physics-aware proxy model (PAPM) that fully incorporates partial prior physics of process systems, which includes multiple input conditions and the general form of conservation relations, resulting in better out-of-sample generalization. Additionally, PAPM contains a holistic temporal-spatial stepping module for flexible adaptation across various process systems. Through systematic comparisons with state-of-the-art pure data-driven and physics-aware models across five two-dimensional benchmarks in nine generalization tasks, PAPM notably achieves an average performance improvement of 6.7%, while requiring fewer FLOPs, and just 1% of the parameters compared to the prior leading method. The code is available at https://github.com/pengwei07/PAPM.

cs.LG↗

arXiv2024

Reference Neural Operators: Learning the Smooth Dependence of Solutions of PDEs on Geometric Deformations

For partial differential equations on domains of arbitrary shapes, existing works of neural operators attempt to learn a mapping from geometries to solutions. It often requires a large dataset of geometry-solution pairs in order to obtain a sufficiently accurate neural operator. However, for many industrial applications, e.g., engineering design optimization, it can be prohibitive to satisfy the requirement since even a single simulation may take hours or days of computation. To address this issue, we propose reference neural operators (RNO), a novel way of implementing neural operators, i.e., to learn the smooth dependence of solutions on geometric deformations. Specifically, given a reference solution, RNO can predict solutions corresponding to arbitrary deformations of the referred geometry. This approach turns out to be much more data efficient. Through extensive experiments, we show that RNO can learn the dependence across various types and different numbers of geometry objects with relatively small datasets. RNO outperforms baseline models in accuracy by a large lead and achieves up to 80% error reduction.

cs.LG↗

arXiv2024

PEAC: Unsupervised Pre-training for Cross-Embodiment Reinforcement Learning

Designing generalizable agents capable of adapting to diverse embodiments has achieved significant attention in Reinforcement Learning (RL), which is critical for deploying RL agents in various real-world applications. Previous Cross-Embodiment RL approaches have focused on transferring knowledge across embodiments within specific tasks. These methods often result in knowledge tightly coupled with those tasks and fail to adequately capture the distinct characteristics of different embodiments. To address this limitation, we introduce the notion of Cross-Embodiment Unsupervised RL (CEURL), which leverages unsupervised learning to enable agents to acquire embodiment-aware and task-agnostic knowledge through online interactions within reward-free environments. We formulate CEURL as a novel Controlled Embodiment Markov Decision Process (CE-MDP) and systematically analyze CEURL's pre-training objectives under CE-MDP. Based on these analyses, we develop a novel algorithm Pre-trained Embodiment-Aware Control (PEAC) for handling CEURL, incorporating an intrinsic reward function specifically designed for cross-embodiment pre-training. PEAC not only provides an intuitive optimization strategy for cross-embodiment pre-training but also can integrate flexibly with existing unsupervised RL methods, facilitating cross-embodiment exploration and skill discovery. Extensive experiments in both simulated (e.g., DMC and Robosuite) and real-world environments (e.g., legged locomotion) demonstrate that PEAC significantly improves adaptation performance and cross-embodiment generalization, demonstrating its effectiveness in overcoming the unique challenges of CEURL. The project page and code are in https://yingchengyang.github.io/ceurl.

cs.LG↗

arXiv2024

DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training

Pre-training has been investigated to improve the efficiency and performance of training neural operators in data-scarce settings. However, it is largely in its infancy due to the inherent complexity and diversity, such as long trajectories, multiple scales and varying dimensions of partial differential equations (PDEs) data. In this paper, we present a new auto-regressive denoising pre-training strategy, which allows for more stable and efficient pre-training on PDE data and generalizes to various downstream tasks. Moreover, by designing a flexible and scalable model architecture based on Fourier attention, we can easily scale up the model for large-scale pre-training. We train our PDE foundation model with up to 0.5B parameters on 10+ PDE datasets with more than 100k trajectories. Extensive experiments show that we achieve SOTA on these benchmarks and validate the strong generalizability of our model to significantly enhance performance on diverse downstream PDE tasks like 3D data. Code is available at \url{https://github.com/thu-ml/DPOT}.

cs.LG↗

arXiv2024

Your Diffusion Model is Secretly a Certifiably Robust Classifier

Generative learning, recognized for its effective modeling of data distributions, offers inherent advantages in handling out-of-distribution instances, especially for enhancing robustness to adversarial attacks. Among these, diffusion classifiers, utilizing powerful diffusion models, have demonstrated superior empirical robustness. However, a comprehensive theoretical understanding of their robustness is still lacking, raising concerns about their vulnerability to stronger future attacks. In this study, we prove that diffusion classifiers possess $O(1)$ Lipschitzness, and establish their certified robustness, demonstrating their inherent resilience. To achieve non-constant Lipschitzness, thereby obtaining much tighter certified robustness, we generalize diffusion classifiers to classify Gaussian-corrupted data. This involves deriving the evidence lower bounds (ELBOs) for these distributions, approximating the likelihood using the ELBO, and calculating classification probabilities via Bayes' theorem. Experimental results show the superior certified robustness of these Noised Diffusion Classifiers (NDCs). Notably, we achieve over 80% and 70% certified robustness on CIFAR-10 under adversarial perturbations with $\ell_2$ norms less than 0.25 and 0.5, respectively, using a single off-the-shelf diffusion model without any additional data.

cs.LG↗

arXiv2024

Preconditioning for Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) have shown promise in solving various partial differential equations (PDEs). However, training pathologies have negatively affected the convergence and prediction accuracy of PINNs, which further limits their practical applications. In this paper, we propose to use condition number as a metric to diagnose and mitigate the pathologies in PINNs. Inspired by classical numerical analysis, where the condition number measures sensitivity and stability, we highlight its pivotal role in the training dynamics of PINNs. We prove theorems to reveal how condition number is related to both the error control and convergence of PINNs. Subsequently, we present an algorithm that leverages preconditioning to improve the condition number. Evaluations of 18 PDE problems showcase the superior performance of our method. Significantly, in 7 of these problems, our method reduces errors by an order of magnitude. These empirical findings verify the critical role of the condition number in PINNs' training.

cs.LG↗

arXiv2024

Accelerating Data Generation for Neural Operators via Krylov Subspace Recycling

Learning neural operators for solving partial differential equations (PDEs) has attracted great attention due to its high inference efficiency. However, training such operators requires generating a substantial amount of labeled data, i.e., PDE problems together with their solutions. The data generation process is exceptionally time-consuming, as it involves solving numerous systems of linear equations to obtain numerical solutions to the PDEs. Many existing methods solve these systems independently without considering their inherent similarities, resulting in extremely redundant computations. To tackle this problem, we propose a novel method, namely Sorting Krylov Recycling (SKR), to boost the efficiency of solving these systems, thus significantly accelerating data generation for neural operators training. To the best of our knowledge, SKR is the first attempt to address the time-consuming nature of data generation for learning neural operators. The working horse of SKR is Krylov subspace recycling, a powerful technique for solving a series of interrelated systems by leveraging their inherent similarities. Specifically, SKR employs a sorting algorithm to arrange these systems in a sequence, where adjacent systems exhibit high similarities. Then it equips a solver with Krylov subspace recycling to solve the systems sequentially instead of independently, thus effectively enhancing the solving efficiency. Both theoretical analysis and extensive experiments demonstrate that SKR can significantly accelerate neural operator data generation, achieving a remarkable speedup of up to 13.9 times.

cs.LG↗

arXiv2023

Improved Operator Learning by Orthogonal Attention

Neural operators, as an efficient surrogate model for learning the solutions of PDEs, have received extensive attention in the field of scientific machine learning. Among them, attention-based neural operators have become one of the mainstreams in related research. However, existing approaches overfit the limited training data due to the considerable number of parameters in the attention mechanism. To address this, we develop an orthogonal attention based on the eigendecomposition of the kernel integral operator and the neural approximation of eigenfunctions. The orthogonalization naturally poses a proper regularization effect on the resulting neural operator, which aids in resisting overfitting and boosting generalization. Experiments on six standard neural operator benchmark datasets comprising both regular and irregular geometries show that our method can outperform competing baselines with decent margins.

cs.LG↗