Source-linked AI summary
Topology Optimization via Machine Learning and Deep Learning: A Review
Seungyeon Shin, Dongju Shin, Namwoo Kang
TL;DR
High computational costs constrain topology optimization despite its ability to design material layouts under load and boundary conditions. This paper reviews ML-based TO from TO and ML perspectives, covering its purposes, methods, limitations, and future directions. The reviewed approaches speed optimization and reduce computational cost, but scalability, data collection, resolution, manufacturability, and computational limitations remain.
Problem
Topology optimization can produce constrained material layouts without a meaningful initial design, but high computational costs limit its use.
Method
The paper reviews MLTO studies from TO and ML perspectives, asking why ML is applied to TO and how TO can be converted into ML problems.
Results
MLTO studies using supervised, unsupervised, and reinforcement learning speed optimization and reduce computational cost across several TO purposes.
Takeaways & Limitations
MLTO research spans acceleration, non-iterative optimization, meta-modeling, design-space reduction, optimizer improvement, generative design, and postprocessing.
Takeaways & Limitations
Current MLTO studies remain limited by scalability, data-collection burden, low topology resolution, manufacturing infeasibility, weak ML performance, and expensive FEA and computation.
Abstract
from arXiv · showhide
Topology optimization (TO) is a method of deriving an optimal design that satisfies a given load and boundary conditions within a design domain. This method enables effective design without initial design, but has been limited in use due to high computational costs. At the same time, machine learning (ML) methodology including deep learning has made great progress in the 21st century, and accordingly, many studies have been conducted to enable effective and rapid optimization by applying ML to TO. Therefore, this study reviews and analyzes previous research on ML-based TO (MLTO). Two different perspectives of MLTO are used to review studies: (1) TO and (2) ML perspectives. The TO perspective addresses "why" to use ML for TO, while the ML perspective addresses "how" to apply ML to TO. In addition, the limitations of current MLTO research and future research directions are examined.
1. Introduction
Topology optimization determines material layouts under load and boundary constraints, but its computational cost limits practical use. This review examines how machine learning and deep learning can accelerate TO, organizing prior work by why ML is used and how TO is formulated as an ML problem.
- Topology optimization: TO determines an optimal material layout within a design space while satisfying load and boundary constraints.The review identifies density-based, evolutionary structural, level-set, and moving morphable component methods as major TO methodologies.
- Computational challenge: 4,137 times: increasing mesh size by 125 times increases required finite element analysis time by this amount.Fine-resolution TO can take hours to days, and three-dimensional TO requires still higher computational costs.
- Computational challenge: Despite GPU, parallel-computing, and other acceleration efforts, TO computational costs remain high and motivate ML-based TO.The review presents MLTO as a response to the persistent computational burden of iterative TO.
- ML-based TO: MLTO applies machine learning or deep learning to quickly and effectively optimize initial topologies.The review focuses especially on methodologies using deep learning.
- Review framework: The review analyzes MLTO from TO and ML perspectives: why ML is applied to TO and how TO can be converted into ML problems.It also examines limitations and future research directions.
2. TO Perspective: Why Use ML
ML is applied to TO to reduce the cost of iterative computation and to enable non-iterative prediction, while also supporting physics-guided and surrogate-based optimization processes. The reviewed studies organize these uses by the purpose they serve in TO, including iteration acceleration, non-iterative optimization, and related prediction or design strategies.
- MLTO purposes: MLTO studies are classified by seven purposes: accelerating iteration, non-iterative optimization, meta-modeling, reducing design space, improving optimizers, generative design, and postprocessing.The classification is presented from the TO perspective, which asks why ML is applied to TO.
- Acceleration of Iteration: Iterative acceleration performs conventional TO through an intermediate stage, then uses ML to predict the optimal topology and avoid the costly convergent stage.This two-stage strategy supplements the later stage, where design changes slowly until convergence.
- Non-Iterative Optimization: Non-iterative MLTO directly derives a topology from design conditions without iterative optimization, but may require multiple networks to improve resolution or engineering performance.Single-stage methods use one network, whereas multistage methods use two or more networks; examples include direct shape prediction and upscaling.
- Single-Stage Process: Physics-guided non-iterative methods incorporate FEA-derived strain, stress, or other physics information as inputs to improve prediction across loading, boundary, and initial-design conditions.Zhang et al. use strain fields with volume fraction and loading or boundary conditions, while TopologyGAN uses strain energy density and von Mises stress.
- Meta-Modeling: ML can replace FEA-based objective or sensitivity calculations, allowing optimization to proceed using predicted compliance, volume fraction, or sensitivity values.CNNs and DNNs have been used for objective prediction, while MLP and SVM-based models have been used for sensitivity prediction.
3. ML Perspective: How to Use ML
The review classifies ML applications to topology optimization by learning algorithm and by how neural models predict, optimize, explore, or postprocess designs. It covers supervised, semi-supervised, unsupervised, and reinforcement learning, alongside architectures such as NNs, CNNs, LSTMs, SVMs, GANs, autoencoders, and modified networks.
- MLTO studies are classified by learning algorithm, input data, loss function, physical-information use, and application generalizability.
- The review compares supervised, unsupervised, semi-supervised, and reinforcement learning according to their treatment of labels, unlabeled data, actions, and rewards.
- GANs and autoencoders are used for design exploration, while reinforcement learning supports optimization and policy-based exploration.
- Supervised TO studies commonly use NNs and CNNs, while also applying LSTMs, SVMs, SVRs, KNNs, random forests, and Gaussian processes.
- NNs and CNNs predict design parameters, objective values, constraints, sensitivities, or optimized shapes for TO.
- CNN-based methods use image representations to predict shape performance or derive optimized designs from loading and boundary conditions.
- LSTMs model sequential TO processes, including predicting later topology sequences and classifying element densities for generative design.
- SVMs support postprocessing by smoothing design boundaries and prediction by estimating sensitivities or identifying checkerboard patterns.
3.2. Data
MLTO models use different data forms according to their purpose: design conditions, intermediate optimization states, final shapes, structural representations, or FEA information. These inputs support prediction, optimization, exploration, postprocessing, and surrogate modeling.
- MLTO input data vary with model purpose, including design parameters, intermediate optimization designs, final shapes, FEA information, and coordinate data.
- Design parameters encode loads, boundary conditions, volume fraction, mass, or coordinates and can be mapped directly to optimal shapes.
- Design-parameter inputs support performance prediction, non-iterative optimization, and generative design exploration.
- Intermediate optimization results are used either to predict objectives, sensitivities, feasibility, or safety, or to accelerate design optimization.
- Final shapes serve prediction, exploration, and postprocessing, including latent-space generation, performance modeling, boundary processing, and resolution enhancement.
- FEA information is primarily used for performance prediction and design optimization, including surrogate sensitivity or gradient calculations.
3.3. Loss Function
The review organizes MLTO loss functions by the training objective and explains three ways FEA information enters deep-learning topology optimization. It also surveys applications, generalizability, and the continuing limitation of 3D expansion.
- MLTO loss functions target shape error, performance error, FEA-derived performance or gradients, additional constraints, or multiple network objectives.
- Shape-based losses compare generated and ground-truth designs using quadratic norms or mean squared error.
- Performance-based losses compare predicted and actual objectives or constraints using measures such as MSE, MAPE, and derivative terms.
- FEA information enters DLTO as performance evaluation during training, as a loss-function term, or as model input.
- Using FEA as evaluation reconstructs training data with high-performing generated designs, whereas embedding it in the loss guides training through backpropagation.
- FEA inputs can combine with design conditions or directly provide data for predicting gradients and sensitivities.
- DLTO applications span beams, bridges, trusses, nanophotonics, metasurfaces, wheels, motors, heat conduction, and fluid structures, with 2D studies still dominant.
- DLTO generalizability is assessed across load and boundary conditions, 3D design problems, and design domains, but expansion to 3D remains limited.
4. Limitations and Research Opportunities
MLTO research faces practical limits despite faster inference and diverse approaches. Key constraints include data and FEA costs, accuracy trade-offs, limited scalability, low-resolution outputs, manufacturability issues, and unstable learning performance.
- Computational and data costs: MLTO can instantly infer an optimal topology after training, but supervised approaches may require extensive optimized training data and costly data generation.Three-dimensional MLTO requires more data, increasing the initial computational burden.
- Computational and data costs: MLTO acceleration trades off against accuracy because iterative TO or FEA inserted into the pipeline increases computational cost and time.In one example, FEA consumed approximately 50% of computational cost for a 60×30 mesh and may become more burdensome at larger scales.
- Physics versus data: Physics-informed neural networks could reduce reliance on large pre-optimized datasets by embedding physics in loss functions, but product-level MLTO results remain un demonstrated.The reviewed PINN work remains limited to benchmark problems despite potential benefits for data efficiency and three-dimensional TO.
- Scalability and industrial applicability: MLTO scalability remains limited because many methods target one design problem, fixed domains, two-dimensional settings, or specific boundary and loading conditions.The review links limited scalability to uncertainty about whether data-collection costs are justified for broader applications.
- Scalability and industrial applicability: Adaptive MLTO may generalize across design domains, boundary conditions, and three-dimensional problems without advance data preparation, although FEA costs remain.This approach reparameterizes the density field with a neural network, but its computational burden still requires improvement.
- Output quality and manufacturing: MLTO outputs often have lower resolution, blurred details, connectivity problems, checkerboard patterns, and may require separate refinement before use.Manufacturing feasibility must also be considered because many topology-optimization results cannot be produced directly by additive manufacturing.
- Learning reliability: MLTO networks require stronger robustness checks because unstable models and cherry-picking can obscure convergence and generated-design quality.GAN-based generative design also lacks established evaluation methods and can suffer from mode collapse.
5. Conclusion
The review finds that ML techniques have been applied across multiple TO tasks to accelerate optimization and reduce computational cost. However, important limitations remain in scalability, data burden, output quality, manufacturability, model performance, and computational expense.
- Conclusion: ML-based TO studies use supervised, unsupervised, and reinforcement learning for seven purposes, including acceleration, non-iterative optimization, generative design, and postprocessing.These approaches were reported to speed optimization and reduce computational cost.
- Conclusion: Current MLTO studies remain limited by scalability across design domains and three-dimensional problems, costly data collection, low-resolution topology, manufacturing infeasibility, weak ML performance, and expensive FEA.The review identifies effective three-dimensional MLTO as an important future research direction.
Appendix A. Background of TO Appendix A introduces the representative TO methodologies that are widely used in ML for TO studies as a foundation.
The appendix frames topology optimization as constrained material-distribution design and introduces representative methods used in MLTO studies. It also uses the MBB half beam to compare four TO methodologies.
- TO formulation: A typical TO problem minimizes an objective function subject to volume and other constraints over a design domain.The material distribution is represented by a density variable that can be void or solid.
- TO formulation: The TO formulation includes a state field satisfying a linear or nonlinear state equation while optimizing the density distribution.The objective and constraints depend on the state field and material distribution.
- Representative methods: The appendix introduces density-based, evolutionary structural optimization, level set, and moving morphable component methods as representative TO approaches.These methods provide a foundation for studies applying ML to topology optimization.
- Representative methods: Figure 24 compares the optimized MBB half beam produced by four topology-optimization methods.The MBB half beam is presented as a representative benchmark problem.
A.1. Density-Based Method
Density-based topology optimization assigns continuous relative densities to elements and optimizes their distribution under stiffness and volume considerations. SIMP, sensitivity analysis, optimality criteria, and filtering are central components of this approach.
- SIMP formulation: SIMP obtains an optimal material distribution for specified load and boundary conditions within a design space.Each element receives a relative density used as the design variable.
- SIMP formulation: The SIMP formulation minimizes compliance while constraining material volume relative to the design-domain volume.Compliance is defined using displacement and stiffness matrices, with density assigned to each element.
- Material interpolation: The Power Law links each element’s Young’s modulus to relative density, solid-material stiffness, void-material stiffness, and a penalty factor.The nonzero minimum stiffness avoids singularity, while the penalty factor reduces the influence of medium-density gray elements.
- Optimization update: Sensitivity analysis computes how changes in element density affect the objective function and constraints during gradient-based optimization.The sensitivities are expressed as differentials with respect to element density under unit element volume.
- Optimization update: The OC method updates design variables and is commonly applied to density-based topology optimization problems.Filtering supplements this process by accounting for neighboring elements to address checkerboard patterns and grayscale issues.
- Filtering: Sensitivity and density filters use weighted averages over neighboring elements within a user-defined mesh-independent radius.Density filtering uses physical density, whereas sensitivity filtering uses sensitivity values and modifies sensitivity analysis accordingly.
A.2. Evolutionary Structural Optimization Method
Evolutionary structural optimization methods derive structures by removing inefficient material, while BESO also adds material where needed. Sensitivity numbers guide these removal and addition decisions, with nodal averaging helping address checkerboard patterns and mesh dependence.
- ESO and BESO: ESO removes unnecessary material from a structure to find an optimal topology, whereas BESO removes unnecessary material and adds it where required.BESO is presented as an advanced extension of ESO.
- Element removal: ESO removes elements according to sensitivity numbers, using thresholds to identify elements below the removal criterion.The sensitivity number is calculated before elements lower than the criterion are removed.
- Sensitivity calculation: BESO uses nodal sensitivity numbers formed from sensitivities of elements connected to each node to address checkerboard patterns.The nodal sensitivity is averaged using element weights.
- Sensitivity calculation: The filtering domain is determined by a radius, and node contributions within that domain are used to calculate improved element sensitivity numbers.The total nodes in the domain and their linear distance-based weights enter the calculation.
- Element removal: Void-element sensitivities are also calculated, and elements below the corresponding deletion threshold are removed.This extends the sensitivity-based decision process beyond solid elements.
A.3. Level Set Method
The level set method represents structural boundaries implicitly through a scalar function and evolves those boundaries during optimization. It produces crisp designs while allowing boundary formation, extinction, and merging of void regions.
- Boundary representation: The level set method captures free structural boundaries and produces crisp designs that require less postprocessing.Unlike conventional shape optimization, it can form, remove, and merge void regions.
- Boundary representation: A scalar level set function represents the boundary as its zero-level curve and permits smooth three-dimensional boundary descriptions.Boundary movement and merging are regulated by physical problems and optimization conditions.
- Boundary evolution: The structural boundary is represented dynamically as the level set function changes over time.The evolving boundary is expressed through a time-dependent set of points satisfying a constant level-set value.
- Boundary evolution: A Hamilton–Jacobi-type equation governs the time-dependent level set function after differentiating the evolving boundary expression.The equation defines the initial-value evolution of the function.
- Optimization coupling: The optimal structural boundary is obtained by solving a PDE whose speed vector depends on the optimization objective.Sensitivity analysis supplies an appropriate value from the objective’s descent direction.
- Optimization coupling: The level set formulation minimizes compliance subject to a volume constraint and the linear elastic equilibrium equation.The objective integrates elastic strain energy over the design domain using the level set indicator.
A.4. MMC Method
The MMC method represents topology using a finite set of movable structural components described by geometric parameters. Components can encode complex topologies compactly while their positions, dimensions, and orientations are optimized under constraints.
- Component representation: MMC organizes structures by optimizing component shapes, lengths, thicknesses, directions, and layouts rather than pixels or nodal points.This parameterization determines the geometric characteristics of the structural components.
- Component representation: MMC can express complex and diverse topologies with a small number of components and allows components to overlap without structural effect in overlapping regions.The method incorporates numerous geometric properties for defined optimization problems.
- Level set encoding: A component’s occupied region can be represented by a level set function distinguishing interior, boundary, and exterior points.The sign of the function identifies whether a coordinate lies inside, on, or outside the component domain.
- Geometric parameters: Changing component position, size, and orientation parameters allows structural components to move, dilate, shrink, and rotate within the design domain.The parameterization includes x0, y0, L, t, and θ.
- Optimization formulation: The MMC topology-optimization problem minimizes an objective over component design variables subject to inequality constraints and admissible parameter sets.Each component parameter vector contains its position, length, thickness, and orientation variables.
Appendix B. Background of ML
Deep learning models are categorized into supervised, unsupervised, semi-supervised, and reinforcement learning according to how they use data and feedback.
- Deep learning models comprise four types: supervised, unsupervised, semi-supervised, and reinforcement learning.The categories distinguish learning from labeled data, unlabeled data, mixtures of both, or interaction-based feedback.
B.1. Supervised Learning
Supervised learning trains models with labeled inputs and outputs, supporting classification and regression tasks. Common supervised algorithms include KNN, SVM, and decision trees.
- Supervised learning trains models using labeled datasets to predict outcomes for new data.
- Classification assigns instances to binary or multiclass categories, whereas regression predicts continuous values from features.
- KNN classifies instances using proximity to similar data points.
- SVM supports classification and regression by maximizing the margin between the separating hyperplane and instances.
- Decision trees sort and classify instances according to their features.
B.2. Unsupervised Learning
Unsupervised learning discovers patterns in unlabeled inputs and is commonly used for clustering, dimensionality reduction, and generative modeling.
- Unsupervised learning receives inputs without labels and learns data characteristics through similarity-based clustering and pattern discovery.
- Clustering: Clustering groups data with similar features, with K-means assigning samples to k clusters using cluster centers and average distances.
- Dimensionality reduction: Dimensionality reduction compresses high-dimensional data into lower-dimensional representations while preserving meaningful variation.
- Generative models: GANs use a generator and discriminator so generated data become similar to reference data, while conditional GANs add conditioning information.
B.3. RL
Reinforcement learning trains an agent through interaction with an environment, using trial and error to choose actions that maximize discounted cumulative rewards.
- Reinforcement learning makes decisions through interaction between an agent and an environment.It is usually based on a Markov decision process but does not require explicit models.
- At time t, the agent takes action a_t and receives state s_t and reward R_t from the environment.
- Trial-and-error training selects actions toward maximizing total reward, with γ ∈ (0,1) discounting future rewards.
- RL frameworks mainly use value-function methods or policy-search methods.