Multiphysics simulation aims to predict and understand interactions between multiple physical phenomena, aiding in comprehending natural processes and guiding engineering design. The system of Partial Differential Equations (PDEs) is crucial for representing these physical fields, and solving these PDEs is fundamental to such simulations. However, current methods primarily yield numerical outputs, limiting interpretability and generalizability. We introduce T-NNGP, a hybrid genetic programming algorithm that integrates traditional numerical methods with deep learning to derive approximate symbolic expressions for multiple unknown functions within a system of PDEs. T-NNGP initially obtains numerical solutions using traditional methods, then generates candidate symbolic expressions via deep reinforcement learning, and finally optimizes these expressions using genetic programming. Furthermore, a universal decoupling strategy guides the search direction and addresses coupling problems, thereby accelerating the search process. Experimental results on three types of PDEs demonstrate that our method can reliably obtain human-understandable symbolic expressions that fit both the PDEs and the numerical solutions from traditional methods. This work advances multiphysics simulation by enhancing our ability to derive approximate symbolic solutions for PDEs, thereby improving our understanding of complex physical phenomena.

Partial differential equations (PDEs) are fundamental in various scientific and engineering fields. Methods based on symbolic regression to solve PDEs have gained attention due to their inherent interpretability. However, existing symbolic regression methods rely solely on genetic programming (GP) during the search process, which presents opportunities for improvement in both precision and stability. We introduce a novel framework, itemd NetGP, which enhances symbolic regression for PDEs in three key aspects. First, NetGP employs prefix notation arrays to represent symbolic expressions, simplifying the evaluation process. Second, to improve the stability of the evolutionary process, deep reinforcement learning is integrated to generate new individuals. Additionally, a novel operator is proposed to avoid the generation of invalid expressions during crossover and mutation of array-based individuals. Empirical evaluations across five types of PDEs demonstrate that NetGP achieves outstanding accuracy and stability in solving these PDEs. The code can be found at https://github.com/grassdeerdeer/NetGP.

Evolutionary multitasking (EMT) is an emerging research direction within the evolutionary computation community, attempting to concurrently solve multiple optimization tasks by exploiting the underlying synergies between the tasks. Recently, numerous explicit transfer strategies have been developed for enhancing positive transfer among optimization tasks. Nevertheless, most of these methods conduct knowledge transfer by transferring the best solutions from a source task to the target task, while ignoring the proper use of information from the target task in solution selection. As a result, the transferred solutions could not well adapt to the target task, thus limiting the effectiveness of knowledge transfer across tasks. To address this issue, this paper proposes a solution selection method based on the lower confidence bound (LCB) for EMT, which is designed by leveraging task-specific information of both source and target tasks. With the proposed LCB metric, a number of high-quality solutions that could be more helpful for the target task can be selected and transferred to enhance positive transfer in EMT. To verify the effectiveness of the proposed approach, the solution selection method is embedded into several existing EMT algorithms and then evaluated on the single-objective multitasking benchmarks, the multiobjective multitasking benchmark, and a real-world application. The obtained results confirmed the generality and efficacy of the proposed solution selection approach.

In recent years, machine learning algorithms, especially deep learning, have shown promising prospects in solving Partial Differential Equations (PDEs). However, as the dimension increases, the relationship and interaction between variables become more complex, and existing methods are difficult to provide fast and interpretable solutions for high-dimensional PDEs. To address this issue, we propose a genetic programming symbolic regression algorithm based on transfer learning and automatic differentiation to solve PDEs. This method uses genetic programming to search for a mathematically understandable expression and combines automatic differentiation to determine whether the search result satisfies the PDE and boundary conditions to be solved. To overcome the problem of slow solution speed caused by large search space, we propose a transfer learning mechanism that transfers the structure of one-dimensional PDE analytical solution to the form of high-dimensional PDE solution. We tested three representative types of PDEs, and the results showed that our proposed method can obtain reliable and human-understandable real solutions or algebraic equivalent solutions of PDEs, and the convergence speed is better than the compared methods. Code of this project is at https://github.com/grassdeerdeer/HD-TLGP.

Neural operators are a class of neural networks to learn mappings between infinite-dimensional function spaces, and recent studies have shown that using neural operators to solve partial differential equations is a promising direction. However, the latest neural operator-based algorithms still leave space for increasing the running speed of algorithms while maintaining accuracy. In this work, we propose a new neural operator by using a neural network to "imitate" the process of the Fourier Transform to solve partial differential equations, called the Imitative Fourier Neural Operator (IFNO). The advantage of this method is to use the property of Fourier Transform to transform the partial differential equation into an algebraic equation, which reduces the fitting difficulty of the neural network and improves the training speed. We compare the proposed algorithm with several state-of-the-art designs on different benchmark instances. The experimental results confirm the effectiveness and performance of the proposed method for solving partial differential equations.

Graph meta-learning has become a preferable paradigm for graph-based node classification with long-tail distribution, owing to its capability of capturing the intrinsic manifold of support and query nodes. Despite the remarkable success, graph meta-learning suffers from severe performance degradation when training on graph data with structural noise. In this work, we observe that the structural noise may impair the smoothness of the intrinsic manifold supporting the support and query nodes, leading to the poor transferable priori of the meta-learner. To address the issue, we propose a new approach for graph meta-learning that is robust against structural noise, called Proxy subgraph-based Manifold Calibration method (Pro-MC). Concretely, a subgraph generator is designed to generate proxy subgraphs that can calibrate the smoothness of the manifold. The proxy subgraph compromises two types of subgraphs with two biases, thus preventing the manifold from being rugged and straightforward. By doing so, our proposed meta-learner can obtain generalizable and transferable prior knowledge. In addition, we provide a theoretical analysis to illustrate the effectiveness of Pro-MC. Experimental results have demonstrated that our approach can achieve state-of-the-art performance under various structural noises.

Rough clustering has attracted increasing attention due to well dealing with the fuzziness and uncertainty of data. It is well known that it needs to manually set the threshold to determine the upper and lower approximations of rough clusters, which may bring a great effect on the clustering performance. When applied to image segmentation, rough clustering is always sensitive to the initialized cluster centers and image noise. Furthermore, only one clustering criterion is considered in rough clustering, which cannot satisfy diverse practical requirements. To handle these issues, a particle competitive mechanism based multiobjective rough clustering algorithm (PCM-MORCA) for image segmentation is proposed. First, a rough intraclass compactness function considering the nonlocal spatial information derived from an image is constructed to overcome the sensitivity to image noise. Next, the constructed rough intraclass compactness function and an interclass separation function are optimized simultaneously to make cluster centers meet diverse segmentation requirements. Then, an adaptive threshold determination mechanism by which the threshold adaptively varies with the clustered data is presented to well determine the upper and lower approximations of rough clusters. After that, to effectively search appropriate cluster centers, a novel pair competition-based particle weight updating strategy is designed for multiobjective particle swarm optimization by improving the elite particle selection and particle update. Finally, a rough clustering index with the nonlocal spatial information is constructed for selecting the optimal solution for PCM-MORCA. Segmentation experiments on Berkeley and magnetic resonance images reveal that PCM-MORCA behaves well on the segmentation accuracy and noise robustness.

We introduce several novel visualization and interaction paradigms for visual analysis of published protein-protein interaction networks, canonical signaling pathway models, and quantitative proteomic data. We evaluate them anecdotally with domain scientists to demonstrate their ability to accelerate the proteomic analysis process. Our results suggest that structuring protein interaction networks around canonical signaling pathway models, exploring pathways globally and locally at the same time, and driving the analysis primarily by the experimental data, all accelerate the understanding of protein pathways. Concrete proteomic discoveries within T-cells, mast cells, and the insulin signaling pathway validate the findings. The aim of the paper is to introduce novel protein network visualization paradigms and anecdotally assess the opportunity of incorporating them into established proteomic applications. We also make available a prototype implementation of our methods, to be used and evaluated by the proteomic community.