Obtaining high-precision aerodynamics in the automotive industry relies on large-scale simulations with computational fluid dynamics, which are generally time-consuming and computationally expensive. Recent advances in operator learning for partial differential equations offer promising improvements in terms of efficiency. However, capturing intricate physical correlations from extensive and varying geometries while balancing large-scale discretization and computational costs remains a significant challenge. To address these issues, we propose **AeroGTO**, an efficient graph-transformer operator designed specifically for learning large-scale aerodynamics in engineering applications. AeroGTO combines local feature extraction through message passing and global correlation capturing via projection-inspired attention, employing a frequency-enhanced graph neural network augmented with k-nearest neighbors to handle three-dimensional (3D) irregular geometries. Moreover, the transformer architecture adeptly manages multi-level dependencies with only linear complexity concerning the number of mesh points, enabling fast inference of the model. Given a car's 3D mesh, AeroGTO accurately predicts surface pressure and estimates drag. In comparisons with five advanced models, AeroGTO is extensively tested on two industry-standard benchmarks, Ahmed-Body and DrivAerNet, achieving a 7.36% improvement in surface pressure prediction and a 10.71% boost in drag coefficient estimation, with fewer FLOPs and only 1% of the parameters used by the prior leading method.

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.

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 **S**orting **K**rylov **R**ecycling (**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.

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.

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.

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 **p**hysics-**a**ware **p**roxy **m**odel (**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.

This work presents orthogonal attention for constructing neural operators to serve as surrogates to model the solutions of a family of Partial Differential Equations (PDEs). The motivation is that the kernel integral operator, which is usually at the core of neural operators, can be reformulated with orthonormal eigenfunctions. Inspired by the success of the neural approximation of eigenfunctions (Deng et al., 2022), we opt to directly parameterize the involved eigenfunctions with flexible neural networks (NNs), based on which the input function is then transformed by the rule of kernel integral. Surprisingly, the resulting NN module bears a striking resemblance to regular attention mechanisms, albeit without softmax. Instead, it incorporates an orthogonalization operation that provides regularization during model training and helps mitigate overfitting, particularly in scenarios with limited data availability. In practice, the orthogonalization operation can be implemented with minimal additional overheads. 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.

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.

While significant progress has been made on Physics-Informed Neural Networks (PINNs), a comprehensive comparison of these methods across a wide range of Partial Differential Equations (PDEs) is still lacking. This study introduces PINNacle, a benchmarking tool designed to fill this gap. PINNacle provides a diverse dataset, comprising over 20 distinct PDEs from various domains, including heat conduction, fluid dynamics, biology, and electromagnetics. These PDEs encapsulate key challenges inherent to real-world problems, such as complex geometry, multi-scale phenomena, nonlinearity, and high dimensionality. PINNacle also offers a user-friendly toolbox, incorporating about 10 state-of-the-art PINN methods for systematic evaluation and comparison. We have conducted extensive experiments with these methods, offering insights into their strengths and weaknesses. In addition to providing a standardized means of assessing performance, PINNacle also offers an in-depth analysis to guide future research, particularly in areas such as domain decomposition methods and loss reweighting for handling multi-scale problems and complex geometry. To the best of our knowledge, it is the largest benchmark with a diverse and comprehensive evaluation that will undoubtedly foster further research in PINNs.

Fourier Neural Operators (FNOs) have shown promise for solving partial differential equations (PDEs). Typically, FNOs employ separate parameters for different frequency modes to specify tunable kernel integrals in Fourier space, which, yet, results in an undesirably large number of parameters when solving high-dimensional PDEs. A workaround is to abandon the frequency modes exceeding a predefined threshold, but this limits the FNOs' ability to represent high-frequency details and poses non-trivial challenges for hyper-parameter specification. To address these, we propose AMortized Fourier Neural Operator (AM-FNO), where an amortized neural parameterization of the kernel function is deployed to accommodate arbitrarily many frequency modes using a fixed number of parameters. We introduce two implementations of AM-FNO, based on the recently developed, appealing Kolmogorov–Arnold Network (KAN) and Multi-Layer Perceptrons (MLPs) equipped with orthogonal embedding functions respectively. We extensively evaluate our method on diverse datasets from various domains and observe up to 31\% average improvement compared to competing neural operator baselines.

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.

Learning partial differential equations’ (PDEs) solution operators is an essential problem in machine learning. However, there are several challenges for learning operators in practical applications like the irregular mesh, multiple input functions, and complexity of the PDEs’ solution. To address these challenges, we propose a general neural operator transformer (GNOT), a scalable and effective transformer-based framework for learning operators. By designing a novel heterogeneous normalized attention layer, our model is highly flexible to handle multiple input functions and irregular meshes. Besides, we introduce a geometric gating mechanism which could be viewed as a soft domain decomposition to solve the multi-scale problems. The large model capacity of the transformer architecture grants our model the possibility to scale to large datasets and practical problems. We conduct extensive experiments on multiple challenging datasets from different domains and achieve a remarkable improvement compared with alternative methods. Our code and data are publicly available at https://github.com/thu-ml/GNOT.

The neural operator has emerged as a powerful tool in learning mappings between function spaces in PDEs. However, when faced with real-world physical data, which are often highly non-uniformly distributed, it is challenging to use mesh-based techniques such as the FFT. To address this, we introduce the Non-Uniform Neural Operator (NUNO), a comprehensive framework designed for efficient operator learning with non-uniform data. Leveraging a K-D tree-based domain decomposition, we transform non-uniform data into uniform grids while effectively controlling interpolation error, thereby paralleling the speed and accuracy of learning from non-uniform data. We conduct extensive experiments on 2D elasticity, (2+1)D channel flow, and a 3D multi-physics heatsink, which, to our knowledge, marks a novel exploration into 3D PDE problems with complex geometries. Our framework has reduced error rates by up to 60% and enhanced training speeds by 2x to 30x. The code is now available at https://github.com/thu-ml/NUNO .

Physics-informed Neural Networks (PINNs) have recently achieved remarkable progress in solving Partial Differential Equations (PDEs) in various fields by minimizing a weighted sum of PDE loss and boundary loss. However, there are several critical challenges in the training of PINNs, including the lack of theoretical frameworks and the imbalance between PDE loss and boundary loss. In this paper, we present an analysis of second-order non-homogeneous PDEs, which are classified into three categories and applicable to various common problems. We also characterize the connections between the training loss and actual error, guaranteeing convergence under mild conditions. The theoretical analysis inspires us to further propose MultiAdam, a scale-invariant optimizer that leverages gradient momentum to parameter-wisely balance the loss terms. Extensive experiment results on multiple problems from different physical domains demonstrate that our MultiAdam solver can improve the predictive accuracy by 1-2 orders of magnitude compared with strong baselines.

Importance sampling (IS) is a popular technique in off-policy evaluation, which re-weights the return of trajectories in the replay buffer to boost sample efficiency. However, training with IS can be unstable and previous attempts to address this issue mainly focus on analyzing the variance of IS. In this paper, we reveal that the instability is also related to a new notion of Reuse Bias of IS --- the bias in off-policy evaluation caused by the reuse of the replay buffer for evaluation and optimization. We theoretically show that the off-policy evaluation and optimization of the current policy with the data from the replay buffer result in an overestimation of the objective, which may cause an erroneous gradient update and degenerate the performance. We further provide a high-probability upper bound of the Reuse Bias and show that controlling one term of the upper bound can control the Reuse Bias by introducing the concept of stability for off-policy algorithms. Based on these analyses, we present a novel yet simple Bias-Regularized Importance Sampling (BIRIS) framework along with practical algorithms, which can alleviate the negative impact of the Reuse Bias, and show that our BIRIS can significantly reduce the Reuse Bias empirically. Moreover, extensive experimental results show that our BIRIS-based methods can significantly improve the sample efficiency on a series of continuous control tasks in MuJoCo.

The design of \emph{de novo} functional proteins that bind with specific ligand molecules is crucial in various domains like therapeutics and bio-engineering. One vital yet challenging step is to design the protein pocket, the cavity region of protein where the ligand binds with. Existing methods suffer from inefficient generation, insufficient context modeling (ligand molecule), and incapability of generating sidechain atoms. To overcome the limitations, we propose a \textbf{F}ull-\textbf{A}tom \textbf{I}terative \textbf{R}efinement framework (\textbf{FAIR}) for protein pocket sequence (i.e., residue types) and 3D structure co-design. Generally, FAIR consists of two steps that follow a coarse-to-fine pipeline (backbone atoms to full atoms including sidechain) for full-atom generation. For efficiency, all residue types and structures are updated together in each round (i.e., full-shot refinement). In the first step, the residue types and backbone coordinates are updated with a hierarchical context encoder and two structure refinement modules capturing inter-residue and pocket-ligand interactions. The second step further models the sidechain atoms of pockets and updates residue types to achieve sequence-structure consistency. The structure of the binding ligand is also updated along with the above refinement iterations accounting for its flexibility. Finally, extensive evaluations show that FAIR outperforms baselines in efficiently designing high-quality pocket sequences and structures. Specifically, the average improvements on AAR and RMSD are over 10$\%$.

Facial expression recognition (FER) has made significant progress over the past few years. But how to overcome the problem of high inter-class similarity and large intra-class difference in FER is still challenging. To address this problem, we propose a novel FER framework called AU-assisted Visual Transformer (AVT) by incorporating facial action units (AU) information into Visual Transformer, which mainly consists of three modules: Local Feature Extraction (LFE) module, Global Relationship Modeling (GRM) module and AU Fusion Module (AFM). Specifically, the LFE module aims to extract local facial expression features by using a deep convolutional neural network, the GRM module is a multi-layer Transformer encoder that captures the relation between local facial regions and obtains a global understanding of the face, and in particular, the AFM introduces fine-grained AU feature and fuses it with expression feature for final classification. Extensive experiments are conducted on RAF-DB and FERPlus datasets, and our AVT achieves competitive results compared to previous state-of-the-art methods, demonstrating the effectiveness of our approach.

Certified defenses such as randomized smoothing have shown promise towards building reliable machine learning systems against $\ell_p$ norm bounded attacks. However, existing methods are insufficient or unable to provably defend against semantic transformations, especially those without closed-form expressions (such as defocus blur and pixelate), which are more common in practice and often unrestricted. To fill up this gap, we propose generalized randomized smoothing (GSmooth), a unified theoretical framework for certifying robustness against general semantic transformations via a novel dimension augmentation strategy. Under the GSmooth framework, we present a scalable algorithm that uses a surrogate image-to-image network to approximate the complex transformation. The surrogate model provides a powerful tool for studying the properties of semantic transformations and certifying robustness. Experimental results on several datasets demonstrate the effectiveness of our approach for robustness certification against multiple kinds of semantic transformations and corruptions, which is not achievable by the alternative baselines.

While deep neural networks have achieved great success in graph analysis, recent work has shown that they are vulnerable to adversarial attacks. Compared with adversarial attacks on image classification, performing adversarial attacks on graphs is more challenging because of the discrete and non-differential nature of the adjacent matrix for a graph. In this work, we propose Cluster Attack --- a Graph Injection Attack (GIA) on node classification, which injects fake nodes into the original graph to degenerate the performance of graph neural networks (GNNs) on certain victim nodes while affecting the other nodes as little as possible. We demonstrate that a GIA problem can be equivalently formulated as a graph clustering problem; thus, the discrete optimization problem of the adjacency matrix can be solved in the context of graph clustering. In particular, we propose to measure the similarity between victim nodes by a metric of Adversarial Vulnerability, which is related to how the victim nodes will be affected by the injected fake node, and to cluster the victim nodes accordingly. Our attack is performed in a practical and unnoticeable query-based black-box manner with only a few nodes on the graphs that can be accessed. Theoretical analysis and extensive experiments demonstrate the effectiveness of our method by fooling the node classifiers with only a small number of queries.

We present a unified hard-constraint framework for solving geometrically complex PDEs with neural networks, where the most commonly used Dirichlet, Neumann, and Robin boundary conditions (BCs) are considered. Specifically, we first introduce the "extra fields'' from the mixed finite element method to reformulate the PDEs so as to equivalently transform the three types of BCs into linear forms. Based on the reformulation, we derive the general solutions of the BCs analytically, which are employed to construct an ansatz that automatically satisfies the BCs. With such a framework, we can train the neural networks without adding extra loss terms and thus efficiently handle geometrically complex PDEs, alleviating the unbalanced competition between the loss terms corresponding to the BCs and PDEs. We theoretically demonstrate that the "extra fields'' can stabilize the training process. Experimental results on real-world geometrically complex PDEs showcase the effectiveness of our method compared with state-of-the-art baselines.

Molecular property prediction (e.g., energy) is an essential problem in chemistry and biology. Unfortunately, many supervised learning methods usually suffer from the problem of scarce labeled molecules in the chemical space, where such property labels are generally obtained by Density Functional Theory (DFT) calculation which is extremely computational costly. An effective solution is to incorporate the unlabeled molecules in a semi-supervised fashion. However, learning semi-supervised representation for large amounts of molecules is challenging, including the joint representation issue of both molecular essence and structure, the conflict between representation and property leaning. Here we propose a novel framework called Active Semi-supervised Graph Neural Network (ASGN) by incorporating both labeled and unlabeled molecules. Specifically, ASGN adopts a teacher-student framework. In the teacher model, we propose a novel semi-supervised learning method to learn general representation that jointly exploits information from molecular structure and molecular distribution. Then in the student model, we target at property prediction task to deal with the learning loss conflict. At last, we proposed a novel active learning strategy in terms of molecular diversities to select informative data during the whole framework learning. We conduct extensive experiments on several public datasets. Experimental results show the remarkable performance of our ASGN framework.