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Partial Differential Equations (PDEs) are notoriously difficult to solve. In general, closed form solutions are not available and numerical approximation schemes are computationally expensive. In this paper, we propose to approach the solution of PDEs based on a novel technique that combines the advantages of two recently emerging machine learning based approaches.
 First, physics-informed neural networks (PINNs) learn continuous solutions of PDEs and can be trained with little to no ground truth data. However, PINNs do not generalize well to unseen domains. Second, convolutional neural networks provide fast inference and generalize but either require large amounts of training data or a physics-constrained loss based on finite differences that can lead to inaccuracies and discretization artifacts.
 We leverage the advantages of both of these approaches by using Hermite spline kernels in order to continuously interpolate a grid-based state representation that can be handled by a CNN. This allows for training without any precomputed training data using a physics-informed loss function only and provides fast, continuous solutions that generalize to unseen domains.
 We demonstrate the potential of our method at the examples of the incompressible Navier-Stokes equation and the damped wave equation. Our models are able to learn several intriguing phenomena such as Karman vortex streets, the Magnus effect, Doppler effect, interference patterns and wave reflections. Our quantitative assessment and an interactive real-time demo show that we are narrowing the gap in accuracy of unsupervised ML based methods to industrial solvers for computational fluid dynamics (CFD) while being orders of magnitude faster.

AAAI 2022

Spline-PINN: Approaching PDEs without Data Using Fast, Physics-Informed Hermite-Spline CNNs

technical paper

The purpose of the AAAI conference is to promote research in artificial intelligence (AI) and scientific exchange among AI researchers, practitioners, scientists, and engineers in affiliated disciplines. 

We study the theoretical properties of random Fourier features classification with Lipschitz continuous loss functions such as support vector machine and logistic regression. Utilizing the regularity condition, we show for the first time that random Fourier features classification can achieve O(1/n^0.5) learning rate with only O(n^0.5) features, as opposed to O(n) features suggested by previous results. Our study covers the standard feature sampling method for which we reduce the number of features required, as well as a problem-dependent sampling method which further reduces the number of features while still keeping the optimal generalization property. Moreover, we prove that the random Fourier features classification can obtain a fast O(1/n) learning rate for both sampling schemes under Massart's low noise assumption. Our results demonstrate the potential effectiveness of random Fourier features approximation in reducing the computational complexity (roughly from O(n^3) in time and O(n^2) in space to O(n^2) and O(n^1.5) respectively) without having to trade-off the statistical prediction accuracy. In addition, the achieved trade-off in our analysis is at least the same as the optimal results in the literature under the worst case scenario and significantly improves the optimal results under benign regularity conditions.

Sharp Analysis of Random Fourier Features in Classification

RGB-NIR fusion is a promising method for low-light imaging. However, high-intensity noise in low-light images amplifies the effect of structure inconsistency between RGB-NIR images, which fails existing algorithms. To handle this, we propose a new RGB-NIR fusion algorithm called Dark Vision Net (DVN) with two technical novelties: Deep Structure and Deep Inconsistency Prior (DIP). The Deep Structure extracts clear structure details in deep multiscale feature space rather than raw input space, which is more robust to noisy inputs. Based on the deep structures from both RGB and NIR domains, we introduce the DIP to leverage the structure inconsistency to guide the fusion of RGB-NIR. Benefits from this, the proposed DVN obtains high-quality low-light images without the visual artifacts. We also propose a new dataset called Dark Vision Dataset (DVD), consisting of aligned RGB-NIR image pairs, as the first public RGB-NIR fusion benchmark. Quantitative and qualitative results on the proposed benchmark show that DVN significantly outperforms other comparison algorithms in PSNR and SSIM, especially in extremely low light conditions.

DarkVisionNet: Low-Light Imaging via RGB-NIR Fusion with Deep Inconsistency Prior

Modern recommender systems are required to adapt to the change in user preferences and item popularity. Such a problem is known as the temporal dynamics problem, and it is one of the main challenges in recommender system modeling. Different from the popular recurrent modeling approach, we propose a new solution named LeapRec to the temporal dynamic problem by using trajectory-based meta-learning to model time dependencies. LeapRec characterizes temporal dynamics by two complement components named global time leap (GTL) and ordered time leap (OTL). By design, GTL learns long-term patterns by finding the shortest learning path across unordered temporal data. Cooperatively, OTL learns short-term patterns by considering the sequential nature of the temporal data. Our experimental results show that LeapRec consistently outperforms the state-of-the-art methods on several datasets and recommendation metrics. Furthermore, we provide an empirical study of the interaction between GTL and OTL, showing the effects of long- and short-term modeling.

Leaping through Time with Gradient-Based Adaptation for Recommendation

Learning to improve AUC performance is an important topic in machine learning. However, AUC maximization algorithms may decrease generalization performance due to the noisy data. Self-paced learning is an effective method for handling noisy data. However, existing self-paced learning methods are limited to pointwise learning, while AUC maximization is a pairwise learning problem. To solve this challenging problem, we innovatively propose a balanced self-paced AUC maximization algorithm (BSPAUC). Specifically, we first provide a statistical objective for self-paced AUC.
 Based on this, we propose our self-paced AUC maximization formulation, where a novel balanced self-paced regularization term is embedded to ensure that the selected positive and negative samples have proper proportions. Specially, the sub-problem with respect to all weight variables may be non-convex in our formulation, while the one is normally convex in existing self-paced problems. To address this, we propose a doubly cyclic block coordinate descent method.
 More importantly, we prove that the sub-problem with respect to all weight variables converges to a stationary point on the basis of closed-form solutions, and our BSPAUC converges to a stationary point of our fixed optimization objective under a mild assumption. Considering both the deep learning and kernel-based implementations, experimental results on several large-scale datasets demonstrate that our BSPAUC has a better generalization performance than existing state-of-the-art AUC maximization methods.

Balanced Self-Paced Learning for AUC Maximization

Ordered Multi-valued Decision Diagram (MDD) is a compact representation used to model various constraints, such as regular constraints and table constraints. It can be particularly useful for representing ad-hoc problem specific constraints. Many algorithms have been proposed to enforce Generalized Arc Consistency (GAC) on MDD constraints. In this paper, we introduce a new compact representation called Binary Constraint Tree (BCT). We propose tree binary encodings to transform any MDD constraint into a BCT constraint. We also present a specialized algorithm enforcing GAC on the BCT constraint resulting from a MDD constraint. Experimental results on a large set of benchmarks show that the BCT GAC algorithm can significantly outperform state-of-the-art MDD as well as table GAC algorithms.

Encoding Multi-Valued Decision Diagram Constraints as Binary Constraint Trees

The complete reason behind a decision is a Boolean formula that characterizes why the decision was made. This recently introduced notion has a number of applications, which include generating explanations, detecting decision bias and evaluating counterfactual queries. Prime implicants of the complete reason are known as sufficient reasons for the decision and they correspond to what is known as PI explanations and abductive explanations. In this paper, we refer to the prime implicates of a complete reason as necessary reasons for the decision. We justify this terminology semantically and show that necessary reasons correspond to what is known as contrastive explanations. We also study the computation of complete reasons for multi-class decision trees and graphs with nominal and numeric features for which we derive efficient, closed-form complete reasons. We further investigate the computation of shortest necessary and sufficient reasons for a broad class of complete reasons, which include the derived closed forms and the complete reasons for Sentential Decision Diagrams (SDDs). We provide an algorithm which can enumerate their shortest necessary reasons in output polynomial time. Enumerating shortest sufficient reasons for this class of complete reasons is hard even for a single reason. For this problem, we provide an algorithm that appears to be quite efficient as we show empirically.

On the Computation of Necessary and Sufficient Explanations

In this paper, a new deep learning network named as graph-based point tracker (GPT) is proposed for 3D object tracking in point clouds. GPT is not based on Siamese network applied to template and search area, but it is based on the transfer of target clue from the template to the search area. GPT is end-to-end trainable. GPT has two new modules: graph feature augmentation (GFA) and improved target clue (ITC) module. The key idea of GFA is to exploit one-to-many relationship between template and search area points using a bipartite graph. In GFA, edge features of the bipartite graph are generated by transferring the target clues of template points to search area points through edge convolution. It captures the relationship between template and search area points effectively from the perspective of geometry and shape of two point clouds. The second module is ITC. The key idea of ITC is to embed the information of the center of the target into the edges of the bipartite graph via Hough voting, strengthening the discriminative power of GFA. Both modules significantly contribute to the improvement of GPT by transferring geometric and shape information including target center from target template to search area effectively. Experiments on the KITTI tracking dataset show that GPT achieves state-of-the-art performance and can run in real-time.

Graph-Based Point Tracker for 3D Object Tracking in Point Clouds

Spiking neural networks (SNNs) can be run on neuromorphic devices with ultra-high speed and ultra-low energy consumption because of their binary and event-driven nature. Therefore, SNNs are expected to have various applications, including as generative models being running on edge devices to create high-quality images. In this study, we build a variational autoencoder (VAE) with SNN to enable image generation. VAE is known for its stability among generative models; recently, its quality advanced. In vanilla VAE, the latent space is represented as a normal distribution, and floating-point calculations are required in sampling. However, this is not possible in SNNs because all features must be binary time series data. Therefore, we constructed the latent space with an autoregressive SNN model, and randomly selected samples from its output to sample the latent variables. This allows the latent variables to follow the Bernoulli process and allows variational learning. Thus, we build the Fully Spiking Variational Autoencoder where all modules are constructed with SNN. To the best of our knowledge, we are the first to build a VAE only with SNN layers. We experimented with several datasets, and confirmed that it can generate images with the same or better quality compared to conventional ANNs. The code is available at https://github.com/kamata1729/FullySpikingVAE.

Fully Spiking Variational Autoencoder

Several methods for discovering interpretable directions in the latent space of pre-trained GANs have been proposed. Latent semantics discovered by unsupervised methods are less disentangled than supervised methods since they do not use pre-trained attribute classifiers. We propose Scale Ranking Estimator (SRE), which is trained using self-supervision. SRE enhances the disentanglement in directions obtained by existing unsupervised disentanglement techniques. These directions are updated to preserve the ordering of variation within each direction in latent space. Qualitative and quantitative evaluation of the discovered directions demonstrates that our proposed method significantly improves disentanglement in various datasets. We also show that the learned SRE can be used to perform Attribute-based image retrieval task without any training.

Self-Supervised Enhancement of Latent Discovery in GANs

Handwritten mathematical expression recognition aims to automatically generate LaTeX sequences from given images. Currently, attention-based encoder-decoder models are widely used in this task. They typically generate target sequences in a left-to-right (L2R) manner, leaving the right-to-left (R2L) contexts unexploited. In this paper, we propose an Attention aggregation based Bi-directional Mutual learning Network (ABM) which consists of one shared encoder and two parallel inverse decoders (L2R and R2L). The two decoders are enhanced via mutual distillation, which involves one-to-one knowledge transfer at each training step, making full use of the complementary information from two inverse directions. Moreover, in order to deal with mathematical symbols in diverse scales, an Attention Aggregation Module (AAM) is proposed to effectively integrate multi-scale coverage attentions. Notably, in the inference phase, given that the model already learns knowledge from two inverse directions, we only use the L2R branch for inference, keeping the original parameter size and inference speed. Extensive experiments demonstrate that our proposed approach achieves the recognition accuracy of 56.85 % on CROHME 2014, 52.92 % on CROHME 2016, and 53.96 % on CROHME 2019 without data augmentation and model ensembling, substantially outperforming the state-of-the-art methods. The source code is available in https://github.com/XH-B/ABM.

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