Efficient Discrete Physics-informed Neural Networks for Addressing Evolutionary Partial Differential Equations

Physics-informed neural networks (PINNs) have shown promising potential for solving partial differential equations (PDEs) using deep learning. However, PINNs face training difficulties for evolutionary PDEs, particularly for dynamical systems whose solutions exhibit multi-scale or turbulent behavior over time. The reason is that PINNs may violate the temporal causality property since all the temporal features in the PINNs loss are trained simultaneously. This paper proposes to use implicit time differencing schemes to enforce temporal causality, and use transfer learning to sequentially update the PINNs in space as surrogates for PDE solutions in different time frames. The evolving PINNs are better able to capture the varying complexities of the evolutionary equations, while only requiring minor updates between adjacent time frames. Our method is theoretically proven to be convergent if the time step is small and each PINN in different time frames is well-trained. In addition, we provide state-of-the-art (SOTA) numerical results for a variety of benchmarks for which existing PINNs formulations may fail or be inefficient. We demonstrate that the proposed method improves the accuracy of PINNs approximation for evolutionary PDEs and improves efficiency by a factor of 4-40x.

Machine Learning Clifford invariants of ADE Coxeter elements

There has been recent interest in novel Clifford geometric invariants of linear transformations. This motivates the investigation of such invariants for a certain type of geometric transformation of interest in the context of root systems, reflection groups, Lie groups and Lie algebras: the Coxeter transformations. We perform exhaustive calculations of all Coxeter transformations for $A_8$, $D_8$ and $E_8$ for a choice of basis of simple roots and compute their invariants, using high-performance computing. This computational algebra paradigm generates a dataset that can then be mined using techniques from data science such as supervised and unsupervised machine learning. In this paper we focus on neural network classification and principal component analysis. Since the output -- the invariants -- is fully determined by the choice of simple roots and the permutation order of the corresponding reflections in the Coxeter element, we expect huge degeneracy in the mapping. This provides the perfect setup for machine learning, and indeed we see that the datasets can be machine learned to very high accuracy. This paper is a pump-priming study in experimental mathematics using Clifford algebras, showing that such Clifford algebraic datasets are amenable to machine learning, and shedding light on relationships between these novel and other well-known geometric invariants and also giving rise to analytic results.

Temporal Consistent Automatic Video Colorization via Semantic Correspondence

Video colorization task has recently attracted wide attention. Recent methods mainly work on the temporal consistency in adjacent frames or frames with small interval. However, it still faces severe challenge of the inconsistency between frames with large interval.To address this issue, we propose a novel video colorization framework, which combines semantic correspondence into automatic video colorization to keep long-range consistency. Firstly, a reference colorization network is designed to automatically colorize the first frame of each video, obtaining a reference image to supervise the following whole colorization process. Such automatically colorized reference image can not only avoid labor-intensive and time-consuming manual selection, but also enhance the similarity between reference and grayscale images. Afterwards, a semantic correspondence network and an image colorization network are introduced to colorize a series of the remaining frames with the help of the reference. Each frame is supervised by both the reference image and the immediately colorized preceding frame to improve both short-range and long-range temporal consistency. Extensive experiments demonstrate that our method outperforms other methods in maintaining temporal consistency both qualitatively and quantitatively. In the NTIRE 2023 Video Colorization Challenge, our method ranks at the 3rd place in Color Distribution Consistency (CDC) Optimization track.

SPColor: Semantic Prior Guided Exemplar-based Image Colorization

Exemplar-based image colorization aims to colorize a target grayscale image based on a color reference image, and the key is to establish accurate pixel-level semantic correspondence between these two images. Previous methods search for correspondence across the entire reference image, and this type of global matching is easy to get mismatch. We summarize the difficulties in two aspects: (1) When the reference image only contains a part of objects related to target image, improper correspondence will be established in unrelated regions. (2) It is prone to get mismatch in regions where the shape or texture of the object is easily confused. To overcome these issues, we propose SPColor, a semantic prior guided exemplar-based image colorization framework. Different from previous methods, SPColor first coarsely classifies pixels of the reference and target images to several pseudo-classes under the guidance of semantic prior, then the correspondences are only established locally between the pixels in the same class via the newly designed semantic prior guided correspondence network. In this way, improper correspondence between different semantic classes is explicitly excluded, and the mismatch is obviously alleviated. Besides, to better reserve the color from reference, a similarity masked perceptual loss is designed. Noting that the carefully designed SPColor utilizes the semantic prior provided by an unsupervised segmentation model, which is free for additional manual semantic annotations. Experiments demonstrate that our model outperforms recent state-of-the-art methods both quantitatively and qualitatively on public dataset.

Exemplar-based Video Colorization with Long-term Spatiotemporal Dependency

Exemplar-based video colorization is an essential technique for applications like old movie restoration. Although recent methods perform well in still scenes or scenes with regular movement, they always lack robustness in moving scenes due to their weak ability in modeling long-term dependency both spatially and temporally, leading to color fading, color discontinuity or other artifacts. To solve this problem, we propose an exemplar-based video colorization framework with long-term spatiotemporal dependency. To enhance the long-term spatial dependency, a parallelized CNN-Transformer block and a double head non-local operation are designed. The proposed CNN-Transformer block can better incorporate long-term spatial dependency with local texture and structural features, and the double head non-local operation further leverages the performance of augmented feature. While for long-term temporal dependency enhancement, we further introduce the novel linkage subnet. The linkage subnet propagate motion information across adjacent frame blocks and help to maintain temporal continuity. Experiments demonstrate that our model outperforms recent state-of-the-art methods both quantitatively and qualitatively. Also, our model can generate more colorful, realistic and stabilized results, especially for scenes where objects change greatly and irregularly.

SCC-rFMQ Learning in Cooperative Markov Games with Continuous Actions

Although many reinforcement learning methods have been proposed for learning the optimal solutions in single-agent continuous-action domains, multiagent coordination domains with continuous actions have received relatively few investigations. In this paper, we propose an independent learner hierarchical method, named Sample Continuous Coordination with recursive Frequency Maximum Q-Value (SCC-rFMQ), which divides the cooperative problem with continuous actions into two layers. The first layer samples a finite set of actions from the continuous action spaces by a re-sampling mechanism with variable exploratory rates, and the second layer evaluates the actions in the sampled action set and updates the policy using a reinforcement learning cooperative method. By constructing cooperative mechanisms at both levels, SCC-rFMQ can handle cooperative problems in continuous action cooperative Markov games effectively. The effectiveness of SCC-rFMQ is experimentally demonstrated on two well-designed games, i.e., a continuous version of the climbing game and a cooperative version of the boat problem. Experimental results show that SCC-rFMQ outperforms other reinforcement learning algorithms.

SA-IGA: A Multiagent Reinforcement Learning Method Towards Socially Optimal Outcomes

In multiagent environments, the capability of learning is important for an agent to behave appropriately in face of unknown opponents and dynamic environment. From the system designer's perspective, it is desirable if the agents can learn to coordinate towards socially optimal outcomes, while also avoiding being exploited by selfish opponents. To this end, we propose a novel gradient ascent based algorithm (SA-IGA) which augments the basic gradient-ascent algorithm by incorporating social awareness into the policy update process. We theoretically analyze the learning dynamics of SA-IGA using dynamical system theory and SA-IGA is shown to have linear dynamics for a wide range of games including symmetric games. The learning dynamics of two representative games (the prisoner's dilemma game and the coordination game) are analyzed in details. Based on the idea of SA-IGA, we further propose a practical multiagent learning algorithm, called SA-PGA, based on Q-learning update rule. Simulation results show that SA-PGA agent can achieve higher social welfare than previous social-optimality oriented Conditional Joint Action Learner (CJAL) and also is robust against individually rational opponents by reaching Nash equilibrium solutions.