Simulating and controlling physical systems described by partial differential equations (PDEs) are crucial tasks across science and engineering. Recently, diffusion generative models have emerged as a competitive class of methods for these tasks due to their ability to capture long-term dependencies and model high-dimensional states. However, diffusion models typically struggle with handling system states with abrupt changes and generalizing to higher resolutions. In this work, we propose Wavelet Diffusion Neural Operator (WDNO), a novel PDE simulation and control framework that enhances the handling of these complexities. WDNO comprises two key innovations. Firstly, WDNO performs diffusion-based generative modeling in the wavelet domain for the entire trajectory to handle abrupt changes and long-term dependencies effectively. Secondly, to address the issue of poor generalization across different resolutions, which is one of the fundamental tasks in modeling physical systems, we introduce multi-resolution training. We validate WDNO on five physical systems, including 1D advection equation, three challenging physical systems with abrupt changes (1D Burgers' equation, 1D compressible Navier-Stokes equation and 2D incompressible fluid), and a real-world dataset ERA5, which demonstrates superior performance on both simulation and control tasks over state-of-the-art methods, with significant improvements in long-term and detail prediction accuracy. Remarkably, in the challenging context of the 2D high-dimensional and indirect control task aimed at reducing smoke leakage, WDNO reduces the leakage by 78% compared to the second-best baseline. The code can be found at https://github.com/AI4Science-WestlakeU/wdno.git.

The application of deep learning for partial differential equation (PDE)-constrained control is gaining increasing attention. However, existing methods rarely consider safety requirements crucial in real-world applications. To address this limitation, we propose Safe Diffusion Models for PDE Control (SafeDiffCon), which introduce the uncertainty quantile as model uncertainty quantification to achieve optimal control under safety constraints through both post-training and inference phases. Firstly, our approach post-trains a pre-trained diffusion model to generate control sequences that better satisfy safety constraints while achieving improved control objectives via a reweighted diffusion loss, which incorporates the uncertainty quantile estimated using conformal prediction. Secondly, during inference, the diffusion model dynamically adjusts both its generation process and parameters through iterative guidance and fine-tuning, conditioned on control targets while simultaneously integrating the estimated uncertainty quantile. We evaluate SafeDiffCon on three control tasks: 1D Burgers’ equation, 2D incompressible fluid, and controlled nuclear fusion problem. Results demonstrate that SafeDiffCon is the only method that satisfies all safety constraints, whereas other classical and deep learning baselines fail. Furthermore, while adhering to safety constraints, SafeDiffCon achieves the best control performance. The code can be found at https://github.com/AI4Science-WestlakeU/safediffcon.

Traditional multiobjective evolutionary algorithms (MOEAs) face challenges when addressing sparse large-scale multiobjective optimization problems (SLSMOPs) with many zero decision variables. The “large-scale” refers to the high dimensionality of the decision space, making it difficult for traditional MOEAs to traverse vast expanses efficiently with limited computational resources. Furthermore, In sparse contexts, most variables in Pareto optimal solutions are zero. It is difficult for traditional MOEAs to identify nonzero variables’ positions efficiently. In reinforcement learning, Thompson sampling employs a probability distribution to estimate each item’s value or success probability. Drawing inspiration from this concept, we propose a Thompson sampling-based sparse evolutionary operator (TSSEO). TSSEO maintains a probability distribution for each variable and utilizes this distribution to recommend for the variable, assisting MOEAs in transitioning from high-dimensionality dense to sparse spaces. Experimental results show that when integrated with representative MOEAs, TSSEO performs competitively in three real-world problems and eight benchmark problems involving up to 10041 decision variables, compared to algorithms designed explicitly for SLSMOPs.

Controlling the evolution of complex physical systems is a fundamental task across science and engineering. Classical techniques suffer from limited applicability or huge computational costs. On the other hand, recent deep learning and reinforcement learning-based approaches often struggle to optimize long-term control sequences under the constraints of system dynamics. In this work, we introduce Diffusion Physical systems Control (DiffPhyCon), a new class of method to address the physical systems control problem. DiffPhyCon excels by simultaneously minimizing both the learned generative energy function and the predefined control objectives across the entire trajectory and control sequence. Thus, it can explore globally and plan near-optimal control sequences. Moreover, we enhance DiffPhyCon with prior reweighting, enabling the discovery of control sequences that significantly deviate from the training distribution. We test our method on three tasks: 1D Burgers' equation, 2D jellyfish movement control, and 2D high-dimensional smoke control, where our generated jellyfish dataset is released as a benchmark for complex physical system control research. Our method outperforms widely applied classical approaches and state-of-the-art deep learning and reinforcement learning methods. Notably, DiffPhyCon unveils an intriguing fast-close-slow-open pattern observed in the jellyfish, aligning with established findings in the field of fluid dynamics. The project website, jellyfish dataset, and code can be found at https://github.com/AI4Science-WestlakeU/diffphycon.

The multimicrogrid network structure design problem (MNSDP) represents a binary matrix optimization challenge, targeting the minimization of the cumulative length of power supply circuits within a multimicrogrid system, subject to specific constraints. The optimization of this problem is pivotal for augmenting the stability and resilience of power systems, particularly in remote locales harnessing renewable energy sources. Given its inherent large-scale, sparse, and multimodal nature, the pursuit of the global optimal solution for MNSDP is inherently complex. In this research, we introduce a sophisticated mathematical model of the MNSDP, accommodating three distinct node types, each having disparate reliability prerequisites. We further unveil a benchmark test suite based on real-world scenarios, dubbed MNSDP-LIB. To further our innovations, we present the large-scale binary matrix-based differential evolution (LBMDE) algorithm. This novel algorithm adopts a binary-matrix-centric DE operator with an enhanced feasibility-centric environmental selection strategy. Empirical experiments accentuate the proficiency of LBMDE in addressing large-scale binary matrix optimization challenges. When juxtaposed with extant evolutionary algorithms and a renowned commercial solver, LBMDE demonstrates commendable competitiveness.

Multimodal multiobjective problems (MMOPs) commonly arise in real-world situations where distant solutions in decision space share a very similar objective value. Traditional multimodal multiobjective evolutionary algorithms (MMEAs) prefer to pursue multiple Pareto solutions that have the same objective values. However, a more practical situation in engineering problems is that one solution is slightly worse than another in terms of objective values, while the solutions are far away in the decision space. In other words, such problems have global and local Pareto fronts (PFs). In this study, we proposed several benchmark problems with several local PFs. Then, we proposed an evolutionary algorithm with a hierarchy ranking method (HREA) to find both the global and the local PFs based on the decision maker’s preference. Regarding HREA, we proposed a local convergence quality evaluation method to better maintain diversity in the decision space. Moreover, a hierarchy ranking method was introduced to update the convergence archive. The experimental results show that HREA is competitive compared with other state-of-the-art MMEAs for solving the chosen benchmark problems.

Multiobjective orienteering problems (MO-OPs) are classical multiobjective routing problems and have received much attention in recent decades. This study seeks to solve MO-OPs through a problem-decomposition framework, that is, an MO-OP is decomposed into a multiobjective knapsack problem (MOKP) and a traveling salesman problem (TSP). The MOKP and TSP are then solved by a multiobjective evolutionary algorithm (MOEA) and a deep reinforcement learning (DRL) method, respectively. While the MOEA module is for selecting cities, the DRL module is for planning a Hamiltonian path for these cities. An iterative use of these two modules drives the population toward the Pareto front of MO-OPs. The effectiveness of the proposed method is compared against NSGA-II and NSGA-III on various types of MO-OP instances. Experimental results show that our method performs best on almost all the test instances and has shown strong generalization ability.

This article introduces a new deep learning approach to approximately solve the covering salesman problem (CSP). In this approach, given the city locations of a CSP as input, a deep neural network model is designed to directly output the solution. It is trained using the deep reinforcement learning without supervision. Specifically, in the model, we apply the multihead attention (MHA) to capture the structural patterns, and design a dynamic embedding to handle the dynamic patterns of the problem. Once the model is trained, it can generalize to various types of CSP tasks (different sizes and topologies) without the need of retraining. Through controlled experiments, the proposed approach shows desirable time complexity: it runs more than 20 times faster than the traditional heuristic solvers with a tiny gap of optimality. Moreover, it significantly outperforms the current state-of-the-art deep learning approaches for combinatorial optimization in the aspect of both training and inference. In comparison with traditional solvers, this approach is highly desirable for most of the challenging tasks in practice that are usually large scale and require quick decisions.

Due to the high variability and uncertainty of the wind speed, an interval forecast can provide more information for decision makers to achieve a better energy management compared to the traditional point forecast. In this article, a knee-based lower upper bound estimation method (K-LUBE) is proposed to construct wind speed prediction intervals (PIs). First, we analyze the underlying limitations of traditional direct interval forecast methods, i.e., their obtained PIs often fail to achieve a good balance between the interval width and the coverage probability. K-LUBE resolves the difficulty based on a multiobjective optimization framework in conjunction with a knee selection criterion. Specifically, a PI-NSGA-II multiobjective optimization algorithm is designed to obtain a set of Pareto-optimal solutions. A parameter transfer and a sample training strategies are developed to significantly improve the convergence speed of the optimization procedure. Then, the knee selection criterion is introduced to select the best tradeoff solution among the obtained solutions. In comparison with traditional methods, this method can always provide a reliable PI for decision makers. The procedure is automatic and requires no parameter to be specified in advance, making it more practical for use. The effectiveness of the proposed K-LUBE method is demonstrated through extensive comparisons with four traditional direct interval forecast methods and four classical benchmark models.

In addition to the search for feasible solutions, the utilization of informative infeasible solutions is important for solving constrained multiobjective optimization problems (CMOPs). However, most of the existing constrained multiobjective evolutionary algorithms (CMOEAs) cannot effectively explore and exploit those solutions and, therefore, exhibit poor performance when facing problems with large infeasible regions. To address the issue, this article proposes a novel method, called DD-CMOEA, which features dual stages (i.e., exploration and exploitation) and dual populations. Specifically, the two populations, called mainPop and auxPop, first individually evolve with and without considering the constraints, responsible for exploring feasible and infeasible solutions, respectively. Then, in the exploitation stage, mainPop provides information about the location of feasible regions, which facilitates auxPop to find and exploit surrounding infeasible solutions. The promising infeasible solutions obtained by auxPop in turn help mainPop converge better toward the Pareto-optimal front. Extensive experiments on three well-known test suites and a real-world case study fully demonstrate that DD-CMOEA is more competitive than five state-of-the-art CMOEAs.

This study contributes to the treatment of numerous constraints in evolutionary many-constraint optimization through consideration of the relationships between pair-wise constraints. In a conflicting relationship, the functional value of one constraint increases as the value in another constraint decreases. In a harmonious relationship, the improvement in one constraint is rewarded with simultaneous improvement in the other constraint. In an independent relationship, the adjustment to one constraint never affects the adjustment to the other. Based on the different features, methods for identifying constraint relationships are investigated, helping to simplify many-constraint optimization problems (MCOPs). Additionally, the transitivity of the relationships is further discussed, facilitating the determination of the relationship in a new pair of constraints.

Evolutionary algorithms (EAs) are often applied to deal with UAV route planning. The solution encoding is one of important factor in designing effective EAs. In a traditional encoding mechanism, each individual represents one route. The whole population then consists of a number of routes. We argue that such an encoding is less effective in route planning, and then proposed an alternative encoding mechanism in which one individual represents only one navigation point. The whole population then represents one route. This implicitly turns EAs into single-point based search with high exploitation ability. To further improve the exploration ability of algorithms using this new encoding, a slightly modified differential evolution operator is applied. Combining the modified DE operator and the new encoding mechanism, the performance of the derived algorithm is significantly improved, obtaining much better route planning results than DE with the traditional encoding mechanism.

This article proposes an end-to-end framework for solving multiobjective optimization problems (MOPs) using deep reinforcement learning (DRL), that we call DRL-based multiobjective optimization algorithm (DRL-MOA). The idea of decomposition is adopted to decompose the MOP into a set of scalar optimization subproblems. Then, each subproblem is modeled as a neural network. Model parameters of all the subproblems are optimized collaboratively according to a neighborhood-based parameter-transfer strategy and the DRL training algorithm. Pareto-optimal solutions can be directly obtained through the trained neural-network models. Specifically, the multiobjective traveling salesman problem (MOTSP) is solved in this article using the DRL-MOA method by modeling the subproblem as a Pointer Network. Extensive experiments have been conducted to study the DRL-MOA and various benchmark methods are compared with it. It is found that once the trained model is available, it can scale to newly encountered problems with no need for retraining the model. The solutions can be directly obtained by a simple forward calculation of the neural network; thereby, no iteration is required and the MOP can be always solved in a reasonable time. The proposed method provides a new way of solving the MOP by means of DRL. It has shown a set of new characteristics, for example, strong generalization ability and fast solving speed in comparison with the existing methods for multiobjective optimizations. The experimental results show the effectiveness and competitiveness of the proposed method in terms of model performance and running time.

Multimodal multiobjective problems (MMOPs) arise frequently in the real world, in which multiple Pareto-optimal solution (PS) sets correspond to the same point on the Pareto front. Traditional multiobjective evolutionary algorithms (MOEAs) show poor performance in solving MMOPs due to a lack of diversity maintenance in the decision space. Thus, recently, many multimodal MOEAs (MMEAs) have been proposed. However, for most existing MMEAs, the convergence performance in the objective space does not meet expectations. In addition, many of them cannot always obtain all equivalent Pareto solution sets. To address these issues, this study proposes an MMEA based on a weighted indicator, termed MMEA-WI. The algorithm integrates the diversity information of solutions in the decision space into an objective space performance indicator to maintain the diversity in the decision space and introduces a convergence archive to ensure a more effective approximation of the Pareto-optimal front (PF). These strategies can readily be applied to other indicator-based MOEAs. The experimental results show that MMEA-WI outperforms some state-of-the-art MMEAs on the chosen benchmark problems in terms of the inverted generational distance (IGD) and IGD in the decision space (IGDX) metrics.

The main challenge in constrained multiobjective optimization problems (CMOPs) is to appropriately balance convergence, diversity and feasibility. Their imbalance can easily cause the failure of a constrained multiobjective evolutionary algorithm (CMOEA) in converging to the Pareto-optimal front with diverse feasible solutions. To address this challenge, we propose a dual-population-based evolutionary algorithm, named c-DPEA, for CMOPs. c-DPEA is a cooperative coevolutionary algorithm which maintains two collaborative and complementary populations, termedPopulation1andPopulation2. In c-DPEA, a novel self-adaptive penalty function, termedsaPF, is designed to preserve competitive infeasible solutions inPopulation1. On the other hand, infeasible solutions inPopulation2are handled using a feasibility-oriented approach. To maintain an appropriate balance between convergence and diversity in c-DPEA, a new adaptive fitness function, namedbCAD, is developed. Extensive experiments on three popular test suites comprehensively validate the design components of c-DPEA. Comparison against six state-of-the-art CMOEAs demonstrates that c-DPEA is significantly superior or comparable to the contender algorithms on most of the test problems.

The evolutionary multi-objective optimization algorithm was used to optimize the clustering and splitting of nodes in the construction of inverted spatial objects index tree(ITSR) in this paper. Considering the objective factors including object's coverage, overlap, group center distance, directory rectangle perimeter and the word similarity between tree nodes, etc., a multi-objective optimization model for solving the optimal ITSR construction is established. To solve the model, this paper find a novel method of constructing high efficiency inverted text-spatial R-tree index. The experimental results show that the algorithm supports the multi-objective optimal clustering of ITSR with high efficiency and accuracy .

The brain storm optimization (BSO) algorithm is a new and promising swarm intelligence paradigm, based on the emerging intelligence of the human brain storming process. However, BSO is ineffective to deal with the large data sets because its clustering and idea updating operations are computationally expensive. Aiming at this issue, we propose a parallelized brain storm optimizer based on Spark framework called PLBSO. The basic idea is to parallelize the complex operations of population clustering and idea updating in BSO so as to reduce the computation cost. Especially, the generation process of new ideas is modified to make BSO more suitable for parallelism. In addition, a new parallelized algorithm called PBAR based on PLBSO is proposed for association rule mining. The performance of PLBSO is evaluated over serialized BSO on complex multimodal benchmarks. Results show that, compared with serialized BSO, PLBSO can acquire a 350% speedup approximately while keeping similar accuracy. Finally, PBAR is adopted to resolve the association rule mining problem on a transactional dataset taken from IBM SPSS modeler. The encouraging results prove the validity of our algorithms.

The brain storm optimization (BSO) algorithm is a new and promising swarm intellgience paradigm, inspired from the behaviors of the human process of brainstorming. The noverty of BSO lies in the clustering mechanism where the ideas are clustered into a set of groups and each idea learns from experiences of one inter-cluster or two intra-cluster neighbors. However, this mechanism is inefficient to deal with complex optimiaiton problems. In this paper, we propose an improved BSO algorithm called OSBSO using orthogonal experimental design (OED) strategy, which aims to discover useful search experiences for improving the convergence and solution accurancy. In OSBSO, two new clustering procedures are developed, i.e., orthogonal initialization and orthogonal clustering. The orthogonal initialization aims to improve the uniformity of the initial cluster centers in the objective space instead of the decision space, which can enhance the convergence performance. The orthogonal clustering uses the information between inter- cluster and intra-cluster indviduals to alleviate the evolution stagnation of clusters. Experiments are conducted on a set of the CEC2017 benchmark functions and the results verify the effectivenss and efficiency of OSBSO.

Decomposition via scalarization is a basic concept for multiobjective optimization. The weighted sum (WS) method, a frequently used scalarizing method in decomposition-based evolutionary multiobjective (EMO) algorithms, has good features such as computationally easy and high search efficiency, compared to other scalarizing methods. However, it is often criticized by the loss of effect on nonconvex problems. This paper seeks to utilize advantages of the WS method, without suffering from its disadvantage, to solve many-objective problems. A novel decomposition-based EMO algorithm called multiobjective evolutionary algorithm based on decomposition LWS (MOEA/D-LWS) is proposed in which the WS method is applied in a local manner. That is, for each search direction, the optimal solution is selected only amongst its neighboring solutions. The neighborhood is defined using a hypercone. The apex angle of a hypervcone is determined automatically in a priori. The effectiveness of MOEA/D-LWS is demonstrated by comparing it against three variants of MOEA/D, i.e., MOEA/D using Chebyshev method, MOEA/D with an adaptive use of WS and Chebyshev method, MOEA/D with a simultaneous use of WS and Chebyshev method, and four state-of-the-art many-objective EMO algorithms, i.e., preference-inspired co-evolutionary algorithm, hypervolume-based evolutionary, θ-dominance-based algorithm, and SPEA2+SDE for the WFG benchmark problems with up to seven conflicting objectives. Experimental results show that MOEA/D-LWS outperforms the comparison algorithms for most of test problems, and is a competitive algorithm for many-objective optimization.

Wind speed prediction plays a crucial role in energy system planning and management. This study proposes to predict wind speed using a cooperative coevolution (CC) genetic algorithm based on back propagation (GABP) neural network. A decomposition method is provided to decompose decision variables into several subcomponents, simultaneously, independence among variables in different subcomponents are kept to a minimum. With neuron-based sub-population (NSP), the simulation results show that the proposed method CC-GABP achieves higher prediction precision.

Decomposition-based algorithms have become increasingly popular for evolutionary multiobjective optimization. However, the effect of scalarizing methods used in these algorithms is still far from being well understood. This paper analyzes a family of frequently used scalarizing methods, the Lpmethods, and shows that the p value is crucial to balance the selective pressure toward the Pareto optimal and the algorithm robustness to Pareto optimal front (PF) geometries. It demonstrates that an Lpmethod that can maximize the search ability of a decomposition-based algorithm exists and guarantees that, given some weight, any solution along the PF can be found. Moreover, a simple yet effective method called Pareto adaptive scalarizing (PaS) approximation is proposed to approximate the optimal p value. In order to demonstrate the effectiveness of PaS, we incorporate PaS into a state-of-the-art decomposition-based algorithm, i.e., multiobjective evolutionary algorithm based on decomposition (MOEA/D), and compare the resultant MOEA/D-PaS with some other MOEA/D variants on a set of problems with different PF geometries and up to seven conflicting objectives. Experimental results demonstrate that the PaS is effective.

Multi-objective evolutionary algorithm based on decomposition (MOEA/D) has become increasingly popular in solving multi-objective problems (MOPs). In MOEA/D, weight vectors are responsible for maintaining a nice distribution of Pareto optimal solutions. Often, we expect to obtain a set of uniformly distributed solutions by applying a set of uniformly distributed weight vectors in MOEA/D. In this paper, we argue that uniformly distributed weights do not produce uniformly distributed solutions, however, uniformly distributed search directions do. Moreover, we propose to perform a pre-organization procedure before running MOEA/D. The procedure matches each weight to its closet candidate solution. Experimental results show (i) MOEA/D with uniformly distributed search directions would exhibit a better diversity performance, and (ii) MOEA/D with the pre-organization procedure performs better, especially for the convergence performance.