Energy-Based Physics-Informed Form Finding for Clustered Tensegrity Structures Jing Qin∗ , Muhao Chen† ∗ Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA
arXiv:2607.12888v1 [cs.LG] 14 Jul 2026
† Department of Mechanical and Aerospace Engineering, University of Houston, Houston, TX 77204, USA
Abstract—Tensegrity form-finding and physical property prediction are fundamental inverse problems in structural mechanics, which aim to determine equilibrium configurations and internal force distributions. These problems are challenging due to strong nonlinearity arising from the coupling between geometry and forces, the need to ensure structural stability, and the enforcement of constraints such as boundary conditions and symmetry. Moreover, traditional methods often lack robustness to noise and outliers. This paper proposes an energy-based learning framework for clustered tensegrity form finding and physical property prediction. The proposed approach incorporates total potential energy minimization and constitutive relations into the training objective, enabling the simultaneous prediction of equilibrium nodal configurations and associated physical quantities, including member forces and force densities. By incorporating energy-based physical losses directly into the learning process, the framework improves physical consistency, robustness, and data efficiency. Numerical experiments on tensegrity structures, including prism and lander systems, show the great potential of the proposed approach and demonstrate its capability for scalable form finding and accurate prediction of structural properties. Index Terms—Clustered tensegrity structures, form finding, physics-informed neural networks, energy-based learning, structural mechanics
I. I NTRODUCTION Tensegrity structures achieve stable load-bearing behavior through self-equilibrated interactions between isolated compression components and a continuous tension network, enabling lightweight and deployable structural designs. Owing to their self-stress equilibrium mechanisms rather than rigid joints, tensegrity systems are typically lightweight, deployable, and mechanically efficient. These properties make tensegrity structures attractive in applications such as aerospace engineering, robotics, and architectural design. A central challenge in the analysis and design of tensegrity systems is the form-finding problem, which aims to identify equilibrium configurations together with the corresponding internal force distributions that satisfy both static equilibrium and structural constraints [1]. From a mathematical perspective, determining an equilibrium tensegrity configuration amounts to solving a nonlinear inverse problem in which structural geometry and internal force distribution must jointly satisfy equilibrium and constitutive constraints. A broad spectrum of computational approaches has been developed to address this challenge. Among the earliest methods, the force density method (FDM) converts ∗ Corresponding author. Email: [email protected].
the equilibrium equations to a linear system by introducing member force densities, leading to fast computation but reduced flexibility for inverse analysis [2]. Dynamic relaxation techniques subsequently introduced pseudo-dynamic iterative procedures for computing equilibrium states and were later extended to accommodate additional structural constraints, including inextensible members [3]. Optimization-based strategies have also been investigated, such as least-squares formulations solved with the Levenberg–Marquardt algorithm [4] and inverse form-finding methods that cast equilibrium recovery as a parameter estimation problem [5]. Recent studies have shown that exploiting the inherent low-rank structure of the force density matrix through optimization techniques can improve the efficiency and accuracy of tensegrity form finding [2]. The rapid growth of simulation and experimental datasets has promoted the adoption of machine learning techniques for tensegrity analysis. In particular, deep neural networks have been utilized to model the nonlinear mapping from structural variables to equilibrium configurations, providing alternatives to conventional numerical solvers [6]. This direction has been further extended to simultaneously estimate both geometric configurations and structural properties within a unified learning framework [7]. To improve physical consistency, physicsinformed neural networks (PINNs) enforce equilibrium equations and geometric constraints during network training, enabling data-free prediction of equilibrium solutions [8]. Related variational and energy-based neural formulations replace pointwise residual minimization with energy minimization principles for mechanics problems [9], [10]. Graph neural networks have also been introduced to exploit the connectivity patterns of tensegrity structures for scalable force and form prediction [11]. Nevertheless, many existing learning methods enforce physics through residual minimization, which may increase computational cost and optimization difficulty while limiting physical consistency. Furthermore, the clustered topology of tensegrity structures remains largely underexploited, motivating reduced-order physics-informed approaches. In this work, we address these limitations by proposing an energy-based physics-informed learning framework for tensegrity form-finding that incorporates clustering and total potential energy minimization. Specifically, clustered tensegrity topology is employed to reduce the number of independent member forces by exploiting intrinsic similarities among structural members. Rather than enforcing governing equations only through residual minimization, our method
learns a parametric map from member rest-length variations to equilibrium nodal configurations by incorporating physics via energy-based loss functions derived from total potential energy minimization and constitutive relations. The resulting governing relations for nodal configurations and internal forces are combined with data-driven loss terms for associated physical quantities, including member forces and force densities. By embedding energy-based loss terms directly into the learning process, the proposed framework improves physical consistency, robustness, and computational efficiency. Moreover, we apply a three-phase optimization strategy. Numerical experiments on tensegrity benchmarks, including prism and lander structures, validate the proposed framework and demonstrate its capability for scalable form finding and accurate prediction of structural properties. The remainder of this paper is organized as follows. Section II introduces the clustering-based reduction for member forces, clustered tensegrity equilibrium equations, and modal information. Then we describe the proposed form-finding learning method based on data, physics, and energy losses, together with a three-phase optimization strategy. Section III presents numerical simulation results for representative tensegrity structures, including the prism and the lander. Finally, we draw conclusions and show future work in Section IV. II. P ROPOSED M ETHOD A. Clustering-Based Reduction for Member Forces Suppose the tensegrity nodal coordinate matrix is given by N = [n1 , . . . , nm ] ∈ R3×m , whose columns are the coordinate vectors of the nodes in the tensegrity. To reduce the representation of member forces, we adopt the clustered string concept in [12], where adjacent strings are treated as a single continuous element passing over joints via pulleys. Assume the number of effective elements in a clustered tensegrity structure is denoted by nec (nec ≤ ne ), including bars, nonclustered strings, and clustered strings. The mapping from the non-clustered tensegrity structure to its clustered counterpart is represented by a clustering matrix Q ∈ Rnec ×ne whose (i, j)th entry is defined as 1, if the jth element belongs to the ith clustered element; qij = (1) 0, otherwise. Then the reduced member force vector tc ∈ Rnec and the full counterpart t ∈ Rne satisfy tc = Qt.
(2)
B. Clustered Tensegrity Equilibrium Equations Consider a tensegrity structure with nn nodes, consisting of na free nodes and nb constrained nodes, where nn = na + nb . In addition, there are ne elements in the structure, among which nc bars and ns strings such that ne = nc + ns . The nodal coordinate vector at the equilibrium state is denoted by n ∈ R3nn , and the free nodal coordinate vector is denoted by na . We use Ea ∈ R3nn ×3na to denote the index matrix that
maps the free nodal indices to the full nodal indices such that na = EaT n. Similarly, we denote the extracting matrix that maps the constrained nodal indices to the full one as Eb . The full nodal coordinates can be recovered via n = Ea na + Eb nb .
(3)
Assume that C ∈ Rne ×nn is a connectivity matrix between tensegrity elements, including bars and strings, and nodes. Let x and l be the respective vectors of member force densities and member lengths. Then the equilibrium conditions of a clustered tensegrity structure are given by the following equations involving the nodal coordinate vector n ∈ R3nn and the clustered member force vector tc ∈ Rnec [13]: EaT Kn = EaT (fex − g),
(4)
EaT At tc = EaT (fex − g),
(5)
where the respective stiffness and the equilibrium matrices are defined as K = (C T diag(x)C) ⊗ I 3 , T
(6) T
A1 = (C ⊗ I3 ) bdiag(N C ) diag(l) T
At = A1 Q .
−1
,
(7) (8)
Here bdiag(·) denotes the block-diagonalization operator that places the columns of a matrix along the block diagonal, while diag(·) denotes the diagonalization operator that places the entries of a vector on the main diagonal. Since the stiffness matrix K depends on the nodal coordinates n, the equilibrium equation (4) is nonlinear in n. Given the prescribed loading and an initial structural configuration, the corresponding equilibrium configuration can be computed using the procedure described in [7, Algorithm 1]. From (4) and (5), we can see that n and t can be treated as a function of the input length ℓ, i.e., the sum of the initial rest length ℓ0 and the change in the rest length dℓ. Theorem 1 (Tensegrity Modal Information). The modal information of a tensegrity structure, including its natural frequencies and vibration modes, is given by solving the following generalized eigenvalue problem: KT aa φ = ω 2 Maa φ,
(9)
where 1 (|C|T diag(m)|C| + ⌊|C|T diag(m)|C|⌋) ⊗ I 3 , 6 Kaa = EaT KT Ea , Maa = EaT M Ea ,
M=
KT = K + At diag(e) diag(a) diag(l)−3 ATt − A1 bdiag(QT tc ) diag(l)−1 AT1 . Here the operator ⌊·⌋ zeros out the off-diagonal entries of a matrix, e and a in Rne denote Young’s modulus and crosssectional area of all elements, respectively. The variable ω represents the natural frequency of the tensegrity system, and φ is the eigenvector corresponding to the mode shape.
The corresponding residual loss is given by
C. Proposed Method The proposed framework integrates supervised learning, variational energy minimization, and constitutive reconstruction within a unified learning architecture. The network is trained to approximate a nonlinear operator that maps from the prescribed member rest-length variations to the corresponding equilibrium configurations and associated physical quantities. Specifically, given the initial rest length vector ℓ0 and the restlength change dℓ, a neural network learns the mapping dℓ 7→ n,
(10)
where n denotes the equilibrium nodal coordinates. Then both the clustered member force vector tc and the member force density vector x can be explicitly expressed in terms of n. Unlike conventional PINNs, which primarily enforce differential equation residuals pointwise, our energy-driven model incorporates physics through energy-based loss functions derived from total potential energy minimization and constitutive relations. This formulation is motivated by the variational nature of tensegrity equilibrium, with stable equilibrium configurations characterized as local minima of the total potential energy. By embedding energy-based physical constraints directly into network training, the proposed approach enhances physical consistency, robustness, and computational efficiency. Let Nθ denote a neural network parameterized by θ. Given the input rest-length change dℓ, the network predicts the equilibrium nodal coordinates b = Nθ (dℓ). n
1 b (lc − l0c )T diag(ec ) diag(ac ) diag(l0c )−1 (b lc − l0c ), 2 (14) and the total potential energy is given by U (b n) =
b. Π(b n) = U (b n) − (fex − g)T n
c ∈ R3×nn is the matrix where 1 is an all-one vector and N b . The clustered member forces are reconstructed form of n through the constitutive relations tbc = diag(ec ) diag(ac ) diag(l0c )−1 (b lc − l0c ),
(12)
where l0c denotes the clustered rest-length vector, and ec and ac are the clustered counterparts of e and a in Theorem 1. The corresponding member force densities are constructed as b = diag(b x l)−1 QT tbc .
(13)
To enforce physical consistency, we incorporate both equilibrium residual loss and energy-based physics loss into the training process. The equilibrium residual associated with the predicted configuration is defined as cn b − (fex − g) , rb = EaT K where c = (C T diag(b K x)C) ⊗ I3 .
(15)
Minimizing Π drives the predicted configurations toward local minima of the total potential energy, thereby enforcing structural equilibrium and constitutive consistency. The overall learning objective comprises both data-driven and physics-informed loss terms. Given N training samples, the data-driven loss for the nodal coordinates reads as N
Ld =
1 X (i) ∥b n − n(i) ∥22 . N i=1
(16)
To further improve the prediction of structural properties, we define the force prediction loss as N
Lf =
1 X b(i) 2 ∥tc − t(i) x(i) − x(i) ∥22 . c ∥2 + ∥b N i=1
(17)
Based on (15), we define the energy-based physics loss as N
Le =
b lc = Qb l,
1 X (i) 2 ∥b r ∥2 . N i=1
In addition, the elastic strain energy of the clustered tensegrity system is defined as
(11)
Using the predicted nodal coordinates, the member lengths and clustered member lengths are computed through the tensegrity geometric relations b cC T )T 1, l = bdiag(N
N
Lr =
1 X Π(b n(i) ). N i=1
(18)
The network parameters are optimized by minimizing a weighted sum of the data-driven and physics-informed losses defined in (16)–(18), i.e., L = λd Ld + λf Lf + λr Lr + λe Le ,
(19)
where λd , λf , λr , and λe are weighting parameters balancing the contributions of the respective loss terms. The present framework simultaneously predicts equilibrium nodal configurations, clustered member forces, and force densities while enforcing energy-based physical consistency. Compared with conventional PINNs based on strong-form residual minimization, our variational formulation naturally incorporates structural stability and constitutive behavior into the learning process, making it particularly suitable for nonlinear tensegrity form-finding problems. Our learning architecture utilizes a multilayer perceptron to approximate the nonlinear mapping from the prescribed restlength variations to the corresponding equilibrium tensegrity configurations. Specifically, let dℓ ∈ Rnec denote the input rest-length variation vector associated with clustered members. The neural network Nθ parameterized by θ predicts the b defined in (11). equilibrium nodal coordinates n The network consists of multiple hidden layers with nonlinear activation functions. Here, we employ the hyperbolic
tangent activation function due to its smoothness and effectiveness for learning nonlinear structural responses. The output layer produces the full equilibrium nodal coordinate vector, from which the corresponding member lengths, clustered member forces, and force densities are reconstructed through the constitutive and geometric relations described previously. To improve training stability and balance the competing objectives between data fitting and physics enforcement, we adopt a three-phase optimization procedure. The proposed staged training strategy gradually introduces force prediction and physics-informed constraints into the learning process. In the first data-driven pretraining phase, the neural network is trained using only the nodal coordinate prediction loss L(1) = λd Ld .
(20)
This initialization stage enables the network to learn the coarse nonlinear mapping from rest-length changes to equilibrium configurations without strong physics constraints. Consequently, the optimization process becomes more stable and is less prone to becoming trapped in poor local minima during the initial training phase. In the second force-assisted learning phase, the force prediction loss is introduced L(2) = λd Ld + λf Lf .
(21)
At this stage, the network simultaneously learns the equilibrium nodal coordinates, clustered member forces, and force densities through the constitutive reconstruction relations. Incorporating force supervision improves the consistency between predicted geometries and internal structural responses. In the final physics-informed optimization phase, both the equilibrium residual loss and the energy-based physics loss are incorporated into the total objective L(3) = λd Ld + λf Lf + λr Lr + λe Le .
(22)
This phase enforces both equilibrium consistency and variational physical admissibility through minimization of the equilibrium residual and the total potential energy. Since equilibrium configurations correspond to stationary points of the total potential energy, our formulation naturally promotes structural stability and physically admissible predictions. Compared with standard PINNs that minimize solely pointwise residual, our hybrid formulation combines data supervision, enforcement of equilibrium residuals, and energy-based physical constraints, leading to improved robustness and prediction accuracy for nonlinear tensegrity form-finding problems. For optimization, we first employ the Adam optimizer for stochastic gradient-based training across all three phases. Adam provides stable convergence during large-scale nonlinear optimization and efficiently handles the multi-term objective functions. After the Adam iterations converge, the network parameters are subsequently improved by the limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) optimizer. By exploiting second-order information, the L-BFGS stage improves prediction accuracy and produces more stable equilibrium predictions near the minimum-energy configurations.
III. S IMULATION R ESULTS In this section, two representative structures with distinct topological complexity, including prism and lander systems, are selected to assess the proposed energy-based physicsinformed framework for tensegrity form-finding and structural property prediction. To evaluate prediction accuracy and reduce randomness, we report the average mean squared error (MSE) over 10 independent trials. Throughout the experiments, the datasets are generated using Algorithm 1 in [7] with different selections of cable restlength changes. In particular, we employ uniformly equidistant sampling of the cable rest-length changes over a prescribed interval. For each example, the generated dataset consists of equilibrium nodal coordinates together with the corresponding clustered member forces and force densities. To investigate the influence of the amount of training data, we consider multiple train-test splitting ratios in the experiments. The datasets are randomly shuffled and partitioned into training and testing sets with training ratios selected from {0.7, 0.75, 0.8, 0.85, 0.9}. During training, all nodal coordinates and physical quantities are normalized using the training set statistics to improve numerical stability and learning performance. The proposed framework simultaneously learns the equilibrium configurations and the associated physical quantities from prescribed rest-length variations via energybased, physics-informed losses and constitutive relations. All experiments are conducted in Python using PyTorch on a desktop workstation with an Intel Core i9-9960X CPU, 64 GB RAM, and Windows 10 Pro. The proposed framework employs a fully connected neural network with five hidden layers, each containing 256 neurons, together with the hyperbolic tangent activation function by default. The network is trained using a three-phase optimization strategy with the Adam optimizer, followed by an L-BFGS refinement stage, as described in Section II. The Adam optimizer is initialized with a learning rate of 10−3 . Unless otherwise specified, the training procedure uses the same network architecture and optimization settings across all experiments, which empirically provide stable convergence and accurate predictions for tensegrity form-finding and structural property prediction tasks. Both bars and strings are modeled as steel (linear elastic) with a mass density of 7,850 kg/m3 , a Young’s modulus of 200 GPa, and a yield strength of 300 MPa. The bars have hollow circular cross-sections with outer and inner radii of 10 mm and 8 mm, respectively, whereas the strings are modeled as solid circular members with a radius of 2 mm. A. Experiment 1: Tensegrity Prism We first evaluate the proposed method on the prism tensegrity structure shown in Fig. 1. A dataset of 1000 equilibrium configurations is generated using [7, Algorithm 1]. For each sample, the inputs are the rest-length variations of the three vertical strings, while the outputs consist of the equilibrium nodal coordinates, clustered member forces, and corresponding member force densities. Specifically, the restlength variations are sampled uniformly from [−0.15, 0] m
Fig. 1. Prism geometry with the three selected strings s1 , s2 , and s3 used as rest-length change inputs. Strings are depicted in red and bars in blue. The prism has a height of h = 0.5 m and a circumradius of r = 0.25 m, defined by the circumscribed circle of the top-node triangle.
order of 10−3 across all training ratios, indicating accurate prediction of the equilibrium configuration even when only 70% of the samples are used for training. As the training ratio increases, the coordinate prediction error generally decreases, which shows that the learned map benefits from additional equilibrium samples while maintaining stable generalization on the test set. The force prediction results in Fig. 3 show a similar trend. Compared with nodal coordinates, member forces are more sensitive to small geometric perturbations because they are determined through the constitutive relations and depend directly on member elongations. The force relative errors remain below approximately 10−1 across all train-test splits, indicating stable prediction of physically consistent internal force states. This demonstrates that the combined use of coordinate supervision, force-density consistency, equilibrium residuals, and total-potential-energy minimization enables the network to recover not only the equilibrium shape but also physically consistent internal force states. B. Experiment 2: Tensegrity Lander The second experiment involves a six-bar tensegrity lander comprising 6 bars and 24 strings depicted in Fig. 4.
Fig. 2. Relative error of predicted coordinates in prism.
for each selected string. The proposed framework learns the nonlinear mapping from the prescribed rest-length variations to the equilibrium structural responses through the energybased physics-informed losses and constitutive relations. The corresponding prediction errors under different training ratios are illustrated in Figs. 2 and 3. As shown in Fig. 2, our approach achieves accurate coordinate prediction for the prism over all tested training ratios. The coordinate relative error remains consistently on the
Fig. 3. Relative error of predicted forces in prism.
Fig. 4. Geometry of the six-bar tensegrity lander. The two strings s1 and s2 are selected as rest-length inputs. Strings are depicted in red and bars in blue. Each bar has a length of 1 m.
Each data sample contains two cable lengths as inputs and 96 output quantities, including the coordinates of 12 unconstrained nodes (36 degrees of freedom), 30 member forces, and 30 nonzero natural frequencies. Consequently, the network takes two input variables and predicts a total of 96 output quantities. Following the data generation procedure of the prism, the rest lengths of two adjacent strings are varied by random perturbations independently and identically drawn from the uniform distribution over the interval [−0.3, 0] m. The prediction errors under various training ratios are shown in Figs. 5 and 6. Compared with the prism, the lander system has a larger output dimension and a more complex threedimensional connectivity pattern, making the learning task more challenging. Even under this more complicated setting, Fig. 5 indicates that the proposed method produces accurate predictions of the free nodal coordinates for all considered training ratios. The error reduction with increasing training data confirms that the network successfully captures the nonlinear mapping between the two cable rest-length changes
Fig. 5. Relative error of predicted coordinates in lander.
ber forces and force densities. Unlike conventional physicsinformed neural networks that primarily enforce pointwise residuals of differential equations, the proposed framework incorporates physics via energy-based loss functions derived from total potential energy minimization and constitutive relations. This variational formulation naturally enforces structural equilibrium and physical consistency during the training. To improve computational efficiency, clustering strategies are incorporated to reduce the number of independent member forces by exploiting intrinsic similarities among structural members. A three-phase optimization strategy is further developed to stabilize the training procedure by progressively introducing geometric, force, and physics-informed constraints. Numerical experiments on prism and lander tensegrity systems illustrate that the proposed framework accurately predicts equilibrium configurations and structural properties across various train-test splitting settings. The results also show improved robustness compared to purely data-driven approaches. The proposed framework offers a scalable variational learning approach for nonlinear tensegrity form-finding and structural analysis. Future work includes experimental validation on physical tensegrity prototypes, symmetry-aware learning, adaptive sampling strategies, and large-scale deployable tensegrity structures. R EFERENCES
Fig. 6. Relative error of predicted forces in lander.
and the resulting equilibrium configuration. Despite the higher output dimension, the proposed method maintains comparable accuracy on the more complex lander structure. Fig. 6 further evaluates the prediction of member forces. The force errors follow the same overall behavior as the coordinate errors, with improved accuracy as more training samples are included. The results indicate that the learned equilibrium configurations are not merely geometrically close to the reference solutions, but also mechanically consistent with the corresponding internal force distribution. This is important for tensegrity form-finding because an accurate shape prediction alone does not guarantee a valid self-stress state. The lander experiment, therefore, verifies that the proposed energy-based physics-informed framework can scale from the relatively simple prism to a higher-dimensional tensegrity structure while preserving both geometric and physical accuracy. IV. C ONCLUSION This paper introduces a variational learning approach for tensegrity form finding that integrates supervised learning with energy minimization and constitutive reconstruction. Specifically, the framework approximates a nonlinear operator that maps prescribed cable rest-length changes to equilibrium configurations together with the corresponding clustered mem-
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