In this paper, we aim to create physical digital twins of deformable objects under interaction. Existing methods focus more on the physical learning of current state modeling, but generalize worse to future prediction. This is because existing methods ignore the intrinsic physical properties of deformable objects, resulting in the limited physical learning in the current state modeling. To address this, we present NeuSpring, a neural spring field for the reconstruction and simulation of deformable objects from videos. Built upon spring-mass models for realistic physical simulation, our method consists of two major innovations: 1) a piecewise topology solution that efficiently models multi-region spring connection topologies using zero-order optimization, which considers the material heterogeneity of real-world objects. 2) a neural spring field that represents spring physical properties across different frames using a canonical coordinate-based neural network, which effectively leverages the spatial associativity of springs for physical learning. Experiments on real-world datasets demonstrate that our NeuSping achieves superior reconstruction and simulation performance for current state modeling and future prediction, with Chamfer distance improved by 20% and 25%, respectively.

In the evolutionary computing community, the remarkable language-handling capabilities and reasoning power of large language models (LLMs) have significantly enhanced the functionality of evolutionary algorithms (EAs), enabling them to tackle optimization problems involving structured language or program code. Although this field is still in its early stages, its impressive potential has led to the development of various LLM-based EAs. To effectively evaluate the performance and practical applicability of these LLM-based EAs, benchmarks with real-world relevance are essential. In this paper, we focus on LLM-based recommender systems (RSs) and introduce a benchmark problem set, named RSBench, specifically designed to assess the performance of LLM-based EAs in recommendation prompt optimization. RSBench emphasizes session-based recommendations, aiming to discover a set of Pareto optimal prompts that guide the recommendation process, providing accurate, diverse, and fair recommendations. We develop three LLM-based EAs based on established EA frameworks and experimentally evaluate their performance using RSBench. Our study offers valuable insights into the application of EAs in LLM-based RSs. Additionally, we explore key components that may influence the overall performance of the RS, providing meaningful guidance for future research on the development of LLM-based EAs in RSs.

Real-world multi-objective optimization problems often rely on physics-based simulators or physical experiments to assess solution quality, resulting in significant computational costs. In such scenarios, Gaussian process (GP) surrogate-assisted optimizers have demonstrated exceptional optimization performance. This article focuses on the theoretical convergence analysis of two existing decomposition-based GP-assisted optimizers: ParEGO with the upper confidence bound (ParEGO-UCB) and ExTrEMO. Unlike prior studies that typically assume a single weight vector for scalarization, this work primarily investigates multi-weight vector settings. Specifically, we analyze the regret bound of ParEGO-UCB within a rigorous theoretical framework and prove that, under a multi-weight vector setting, its convergence rate surpasses that of GP-UCB, which independently and sequentially optimizes multiple decomposed subproblems. Building on this foundation, we further explore the convergence properties of ExTrEMO, an expensive multi-objective optimizer designed for multi-source transfer optimization, in the context of multi-weight vector settings. Theoretical findings reveal that ExTrEMO achieves a tighter regret bound in multi-source settings compared to single-source scenarios, highlighting the advantages of leveraging additional sources to enhance optimization efficiency and convergence.

Text-to-3D shape generation has shown great promise in generating novel 3D content based on given text prompts. However, existing generative methods mainly consider geometric or visual plausibility while ignoring functionality for the generated 3D shapes. This greatly hinders the practicality of generated 3D shapes in real-world applications. Towards physical AI, we propose Fun3D, a physics-compliant functional text-to-3D shape generation method. By analyzing the solid mechanics of generated 3D shapes, we reveal that the 3D shapes generated by existing text-to-3D generation methods are impractical for real-world applications, as the generated 3D shapes do not comply with the physical laws. To this end, we leverage 3D diffusion models to provide 3D shape priors and design a data-driven differentiable physics layer to optimize 3D shape priors with solid mechanics. This allows us to optimize geometry efficiently and learn physical information about 3D shapes at the same time. Experimental results demonstrate that our method can consider both geometric plausibility and functional requirement, further bridging 3D virtual modeling and physical worlds to advance physical AI.

Transfer multiobjective optimization promises sample-efficient discovery of near Pareto-optimal solutions to a target task by utilizing experiential priors from related source tasks. In this paper, we show that in domains where evaluation data is at a premium, e.g., in scientific and engineering disciplines involving time-consuming computer simulations or complex real-world experimentation, knowledge transfer through surrogate models can be pivotal in saving sample evaluation costs. While state-of-the-art algorithms (without transfer) typically assume budgets in the order of only a few hundred evaluations, we seek to explore how far we can get on even tighter budgets. The uniqueness of our proposed Expensive Transfer Evolutionary Multiobjective Optimizer (ExTrEMO) is that it can maximally utilize external information from hundreds of source datasets, including those that may be negatively correlated with the target task. This is achieved by melding evolutionary search with factorized transfer Gaussian process surrogates, capturing varied source-target correlations in potentially decentralized computation environments. We provide a regret bound analysis for ExTrEMO that translates to a theoretical proof of increasingly faster convergence as a result of multi-source transfers. The theory is experimentally verified on benchmark functions and toward accelerated design of biomedical microdevices. We release our code at https://github.com/LiuJ-2023/ExTrEMO.

This article proposes a constrained evolutionary Bayesian optimization (CEBO) algorithm to cope with expensive constrained optimization problems with inequality constraints. The uniqueness of CEBO lies in its capability of balancing feasibility and objective improvement under a limited function evaluation budget, which is achieved by designing two strategies to obtain promising solutions. The first strategy prefers feasibility. It tends to obtain a feasible solution by utilizing the predicted value and uncertainty provided by Gaussian process (GP). The second strategy prefers objective improvement. It maintains and evolves the population of evolutionary algorithms, and selects a solution with a good objective function value and violating the constraints not too much based on the predicted value and uncertainty provided by GP at each iteration. The sequential implementation of these two strategies allows CEBO to balance feasibility and objective improvement. The effectiveness of CEBO is verified by 26 test instances and a practical application. The results demonstrate that CEBO is able to find high-quality solutions with 100 FEs.

Recently, evolutionary multitasking has been employed to generate a “set of Pareto sets” (SOS) for machine learning models, addressing diverse task settings across heterogeneous environments. This involves creating a repository of compact, specialized solution models that are collectively tailored to each specific task setting and environment, enabling users to select the most suitable model based on particular specifications and preferences. In this paper, we further demonstrate the versatility and applicability of the SOS concept across diverse domains, focusing on three real-world problems: engineering design problems, inventory management problems, and hyperparameter optimization problems. Additionally, as evolutionary multitasking has proven effective in generating the SOS, we investigate the performance of current evolutionary multitasking methods on these real-world problems. Subsequently, we present visualizations of the generated SOS in both decision and objective spaces, complemented by the development of a measurement to gauge the similarity between different Pareto sets corresponding to diverse tasks. Finally, we show that by systematically examining the shifts in Pareto optimal designs across different task settings though the SOS solutions, users can gain deeper understandings on the dynamic interplay between design solutions and their performance in different settings or contexts.

Transfer optimization enables data-efficient optimization of a target task by leveraging experiential priors from related source tasks. This is especially useful in multiobjective optimization settings where a set of tradeoff solutions is sought under tight evaluation budgets. In this article, we introduce a novel concept of inverse transfer in multiobjective optimization. Inverse transfer stands out by employing Bayesian inverse Gaussian process models to map performance vectors in the objective space to population search distributions in task-specific decision space, facilitating knowledge transfer through objective space unification . Building upon this idea, we introduce the first Inverse Transfer Evolutionary Multiobjective Optimizer (invTrEMO). A key highlight of invTrEMO is its ability to harness the common objective functions prevalent in many application areas, even when decision spaces do not precisely align between tasks. This allows invTrEMO to uniquely and effectively utilize information from heterogeneous source tasks as well. Furthermore, invTrEMO yields high-precision inverse models as a significant byproduct, enabling the generation of tailored solutions on-demand based on user preferences. Empirical studies on multi- and many-objective benchmark problems, as well as a practical case study, showcase the faster convergence rate and modeling accuracy of the invTrEMO relative to state-of-the-art evolutionary and Bayesian optimization algorithms. The source code of the invTrEMO is made available at https://github.com/LiuJ-2023/invTrEMO .

This article devises a two-phase Kriging-assisted evolutionary algorithm (named TEA) to tackle expensive constrained multiobjective optimization problems (CMOPs). In the first phase, only objectives are considered, which can help the population to cross infeasible obstacles and to evolve toward the unconstrained Pareto front. Since the unconstrained Pareto front is in front of the feasible region in the objective space, the first phase can find some feasible solutions during the evolution. In the second phase, both objectives and constraints are considered. In this article, we also propose two transition conditions to judge whether the search should be switched from the first phase to the second phase, by making use of the candidates evaluated by the original objectives and constraints in the first phase. These two transition conditions aim at maintaining some high-quality feasible solutions when the first phase ends, which is able to motivate the population to converge toward the constrained Pareto front with good diversity in the second phase. Furthermore, in both phases, we design a new Pareto dominance relationship (called PDPD) by incorporating the probability distribution information derived from the Kriging models. PDPD is further generalized to handle constraints in expensive CMOPs, Constrained PDPD (CPDPD), which provides high credibility for the comparison between two individuals with respect to both objectives and constraints. Finally, three benchmark test suites and a real-world application confirm the superiority of TEA.

Although many methods have been proposed to solve expensive optimization problems (EOPs), they often consume hundreds of function evaluations (FEs) to find the optimal solution, which is unacceptable when facing highly EOPs. To reduce the number of FEs, we incorporate the population distribution into the well-known expected improvement (EI); thus, a new infill criterion called evolutionary EI (EEI) is proposed. In EEI, the covariance matrix adaptation evolution strategy is used to provide the population distribution. Compared with the original EI, EEI focuses more on promising regions provided by the population distribution, thus, reducing the FEs wasted in unpromising regions. By employing EEI as the infill criterion of Bayesian optimization, a new algorithm called EEI-BO is designed. Moreover, we also introduce an extended version of EEI-BO, called EEI-BO+, to handle multitask EOPs. To verify the effectiveness of EEI-BO, it is used to solve 10−, 20−, and 30-D test problems by using only 40, 50, and 60 FEs, respectively. The results show that EEI-BO is able to obtain high-quality solutions by consuming limited FEs. In addition, we apply EEI-BO to deal with the lightweight and crashworthiness design of the side body of an automobile. The results demonstrate that EEI-BO performs well on solving it. Furthermore, the performance of EEI-BO+is investigated by nine test problems. The results show that it has the capability to solve multitask EOPs with fast convergence speed.

A mixed-integer programming (MIP) problem contains both constraints and integer restrictions. Integer restrictions divide the feasible region defined by constraints into multiple discontinuous feasible parts. In particular, the number of discontinuous feasible parts will drastically increase with the increase of the number of integer decision variables and/or the size of the candidate set of each integer decision variable. Due to the fact that the optimal solution is located in one of the discontinuous feasible parts, it is a challenging task to solve a MIP problem. This article presents a cutting and repulsion-based evolutionary framework (called CaR) to solve MIP problems. CaR includes two main strategies: 1) the cutting strategy and 2) the repulsion strategy. In the cutting strategy, an additional constraint is constructed based on the objective function value of the best individual found so far, the aim of which is to continuously cut unpromising discontinuous feasible parts. As a result, the probability of the population entering a wrong discontinuous feasible part can be decreased. In addition, in the repulsion strategy, once it has been detected that the population has converged to a discontinuous feasible part, the population will be reinitialized. Moreover, a repulsion function is designed to repulse the previously explored discontinuous feasible parts. Overall, the cutting strategy can significantly reduce the number of discontinuous feasible parts and the repulsion strategy can probe the remaining discontinuous feasible parts. Sixteen test problems developed in this article and two real-world cases are used to verify the effectiveness of CaR. The results demonstrate that CaR performs well in solving MIP problems.

As an effective optimization tool for expensive optimization problems (EOPs), surrogate-assisted evolutionary algorithms (SAEAs) have been widely studied in recent years. However, most current SAEAs are designed for continuous/ combinatorial EOPs, which are not suitable for mixed-variable EOPs. This article focuses on one kind of mixed-variable EOP: EOPs with continuous and categorical variables (EOPCCVs). A multisurrogate-assisted ant colony optimization algorithm (MiSACO) is proposed to solve EOPCCVs. MiSACO contains two main strategies: 1) multisurrogate-assisted selection and 2) surrogate-assisted local search. In the former, the radial basis function (RBF) and least-squares boosting tree (LSBT) are employed as the surrogate models. Afterward, three selection operators (i.e., RBF-based selection, LSBT-based selection, and random selection) are devised to select three solutions from the offspring solutions generated by ACO, with the aim of coping with different types of EOPCCVs robustly and preventing the algorithm from being misled by inaccurate surrogate models. In the latter, sequence quadratic optimization, coupled with RBF, is utilized to refine the continuous variables of the best solution found so far. By combining these two strategies, MiSACO can solve EOPCCVs with limited function evaluations. Three sets of test problems and two real-world cases are used to verify the effectiveness of MiSACO. The results demonstrate that MiSACO performs well in solving EOPCCVs.

A considerable number of surrogate-assisted evolutionary algorithms (SAEAs) have been developed to solve expensive optimization problems (EOPs) with continuous objective functions. However, in the real-world applications, we may face EOPs with discontinuous objective functions, which are also called EOPs with discontinuous responses (EOPDRs). Indeed, EOPDRs pose a great challenge to current SAEAs. In this article, a surrogate-assisted differential evolution (DE) algorithm with region division is proposed, named ReDSADE. ReDSADE includes three main strategies: 1) the region division strategy; 2) the Kriging-based search; and 3) the radial basis function (RBF)-based local search. In the region division strategy, we define a new distance measure, called the objective-decision distance. Based on this distance, the evaluated solutions are partitioned into several clusters, and several support vector machine (SVM) classifiers are trained to classify them. These SVM classifiers divide the decision space into several subregions, with the aim of making the objective function continuous in them. In the Kriging-based search, a Kriging model is established in each subregion and combined with DE to search for the optimal solution. In the RBF-based local search, DE is coupled with RBF to search around the best solution found so far, thus accelerating the convergence. By combining these three strategies, ReDSADE is able to solve EOPDRs with limited function evaluations. Three sets of test problems and a real-world application are utilized to verify the effectiveness of ReDSADE. The results demonstrate that ReDSADE exhibits good convergence accuracy and convergence speed.

A mixed-integer programming (MIP) problem contains not only constraints but also integer restrictions. Integer restrictions divide the feasible region defined by constraints into multiple discontinuous feasible parts with different sizes. Several popular methods (e.g., rounding and truncation) have been proposed to deal with integer restrictions. Although it is easy for these methods to generate an integer, they tend to converge to an integer which is located in a feasible part with a big size. If the optimal solution is not in this feasible part, they are very likely to converge to a local optimal solution due to the loss of diversity of the population. To overcome this shortcoming, a biobjective optimization-based two-phase method is proposed in this article. In the first phase, a measure function is designed to compute the degree that a solution violates integer restrictions. By employing this measure function as the second objective function and removing integer restrictions, a MIP problem is transformed into a constrained biobjective optimization problem (CBOP). It can be proven that the Pareto optimal solution of the transformed CBOP which satisfies integer restrictions is the optimal solution of the original MIP problem. To solve the transformed CBOP, a new comparison rule is designed. After the first phase, the population can approach the Pareto optimal solution which satisfies integer restrictions. Then, the second phase is implemented to enhance the convergence precision and obtain the optimal solution. In addition, we design 12 test problems to verify the effectiveness of the proposed method. The results demonstrate that the proposed method shows better performance against five state-of-the-art evolutionary algorithms for MIP.