Real-time design of architectural structures with differentiable mechanics and neural networks
Authors: Rafael Pastrana
Eder Medina
Isabel M. de Oliveira
Sigrid Adriaenssens
Ryan P Adams
Venue: ICLR 2025 Poster (2025)
Keywords: Differentiable physics
mechanical design
physics-in-the-loop neural networks
inverse problems
architectural structures
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: We couple neural networks with a differentiable mechanics simulator to accelerate the solution of shape-matching problems for mechanical design.

Abstract: Designing mechanically efficient geometry for architectural structures like shells, towers, and bridges, is an expensive iterative process.
Existing techniques for solving such inverse problems rely on traditional optimization methods, which are slow and computationally expensive, limiting iteration speed and design exploration.
Neural networks would seem to offer a solution via data-driven amortized optimization, but they often require extensive fine-tuning and cannot ensure that important design criteria, such as mechanical integrity, are met.
In this work, we combine neural networks with a differentiable mechanics simulator to develop a model that accelerates the solution of shape approximation problems for architectural structures represented as bar systems.
This model explicitly guarantees compliance with mechanical constraints while generating designs that closely match target geometries.
We validate our approach in two tasks, the design of masonry shells and cable-net towers.
Our model achieves better accuracy and generalization than fully neural alternatives, and comparable accuracy to direct optimization but in real time, enabling fast and reliable design exploration.
We further demonstrate its advantages by integrating it into 3D modeling software and fabricating a physical prototype.
Our work opens up new opportunities for accelerated mechanical design enhanced by neural networks for the built environment.

Published as a conference paper at ICLR 2025
REAL-TIME DESIGN OF ARCHITECTURAL STRUCTURES WITH DIFFERENTIABLE MECHANICS AND NEURAL NETWORKS
Rafael Pastrana, Eder Medina, Isabel M. de Oliveira,"
Sigrid Adriaenssens," 3 Ryan P. Adams
'Architectrre *Computer Science Princeton University
SCivil and Environmental Engincering
(a Carpastrana, em2368, imdo, sadriaen, rpa) @princeton.ede edu
ABSTRACT
Designing mechanically efficient geometry for architectural structures like shells, towers, and bridges, an expensive iterative process. Existing techniques for solv- ing such inverse problems rely on traditional optimization methods, which are slow and computationally expensive, limiting iteration speed and design exploration. Neural networks would seem to offer a solution via data-driven amortized optimiza- tion, but they often require extensive fine-tuning and cannot ensure that important design criteria, such as mechanical integrity, are met. In this work, we combine neural networks with a differentiablee mechanics simulator to develop a model that accelerates the solution of shape approximation problems for architectural structures represented as bar systems. This model explicitly guarantees compliance with mechanical constraints while generating designs that closely match target geometries. We validate our approach two tasks, the design of masonry shells and cable-net towers. Our model achieves better accuracy and generalization than fully neural alternatives, and comparable accuracy to direct optimization but in real time, enabling fast and reliable design exploration. We further demonstrate its ad advantages by integrating it into 3D modeling software and fabricating physical prototype. Our work opens up new opportunities for accelerated mechanical design enhanced by neural networks for the built environment.
INTRODUCTION
Mechanical efficiency is required for architectural structures to span hundreds meters under extreme loads safely with low material volume. An efficient structure sustains loads with small physical element sizes compared to its span, such as thin bars or slender plates, thus minimizing its material footprint. Additionally, shells, towers, and bridges-example of such systems must comply with geometric constraints arising from architecture and fabrication requirements to become feasible structures in the built environment. Designing shapes for such long-span structures, which must fulfill mechanical efficiency and geometric constraints, is complex task requiring substantial domain expertise and human effort. Our goal is to use machine learning to accelerate this challenging task without compromising safety-critical aspects of the design.
One way to approach this problem is start from the mechanical standpoint, employing special- ized mechanical model that directly computes efficient geometry for structures modeled as bar systems (Bletzinger and Ramm, 2001; Bletzinger et al., 2005). Unlike standard, finite element-based mechanical analysis, where one first defines the structure's geometry and then obtains its internal forces, these specialized models-known as form-finding methods in structural engineering (Veenen- daal and Block, 2012; Adriaenssens et al., 2014)-revee reverse the relationship between geometry and force to produce mechanically efficient shapes a single forward solve (Shin et al., 2016). As result, these methods have been successfully applied design landmark structures with thickness-to-s sp-s ratios up to 1:70 (less than that of an eggshell), across a wide palette materials, including stainless steel (Schlaich, 2018), reinforced concrete (Isler, 1994), and stone (Block et al., 2017).

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Published as a conference paper at ICLR 2025
Target Shape goal
Encoder Neural network
Decoder Sinulator
Prediction constrainod
Mechanical design Masoury shell
q
X
b
Figure I: Architecture of our model to amortize the generation of mechanically efficient geometry. A neural network first maps target shape X sampled from family of shapes X' to a stiffness space q. The stiffnesses, in tandem with the boundary conditions b, are then decoded by a mechanical simulator into a physics- chostcc--onttrainee shape X that approximates the target. The predicted shape can be used as the base geometry to design efficient structures like gridshells or masonry shells.
While utiliizig specialized model can lead to efficient designs, it is difficult to guide solutions toward particular geometries as the designer only has explicit control of the mechanical behavior and not the shape. To solve this inverse problem, form-finding is complemented with optimization algorithms to find internal force states that satisfy geometric goals (Panozzo et al., 2013; Maia Avelino et al., 2021). For a designer, however, is cumbersome to perform a time-consuming and computationally intensive optimization when exploring shapes. Practical design requires the evaluation of several target shapes to align with geometric desiderata, multiplying computational effort as each shape requires its own optimization. Therefore, using specialized mechanical models and optimization together is effective but inefficient in practice, where shape variety and real-time feedback are essential.
Neural networks (NNs) can accelerate the mechanical design of architectural structures with data- driven surrogate models that amortize inverse problems, enabling more responsive design tools. Recent applications include the design of tall buildings (Chang and Cheng. 2020), truss lattices (Bastek et al., 2022), and reticulated shells (Tam, 2023). Nevertheless, these purely data-driven examples require learning representations of the inverse problem and the underlying physics at the same time. Even with physics-infone informed neural networks (PINNs) (Raissi et al., 2019b; Karniadakis al., 2021; Lu et al., 2021: Bastek and Kochmann, 2023) trained with sophisticated loss balancing schemes (Bischof and Kraus, 2021; Wang et al., 2022), there is no guarantee of mechanical integrity in their predictions. Here, define integrity the accuracy in predicting the mechanical response of a structure by respecting physics laws. Assurance of mechanical integrity is foundational tenet in structural engineering, where poor predictions might lead to catastrophic collapse and the loss of human lives. In contrast, mechanical simulators in structural engineering have been developed for decades and offer a principled and interpretable way to represent the physics of long-span structures. A hybrid solution would seem ideal, in which neural network amortization is integrated with differentiable physics models (Belbute-Pere Perres et al., 2020; Um al., 2021; Thuerey et al., 2022: Oktay al., 2023) to construct class of machine learning models that shift the current paradigm from a physics-informec informed to a physics in- in-the-loop approach for safety-critical applications.
In this paper, develop neural surrogate model that couples neural network with a differentiable mechanics simulator to enable the solution of shape approximation problems for architectural structures in real time (Figure 1). The coupled model offers advantages over direct gradient- based optimization and current fully neural alternatives for interactive mechanical design. We evaluate our method in two design tasks of increasing complexity: masonry shells and cable-net towers. Our contributions are threefold. First, we demonstrate that our model generates mechanically sound predictions at higher accuracy than NNs and PINNs of similar architecture. Our model exhibits better generalization performance than an equivalent PINN in the masonry shells task. Second, we show that our model reaches comparable accuracy to optimization, but it is up to four orders of magnitude faster. In the cable-nets task, our model provides robust initialization for direct optimization, outperforming the designs generated by optimization initialized with human domain expertise. Third, we showcase the application of a maturing machine learning technique (i.e., coupling learnable and analytical components in the same architecture) to a new, high-impact domain for physical design (i.e., architectural structures). To illustrate its practical impact, we deploy trained model in a 3D modeling program to design a shell and then fabricate a physical prototype of the predicted geometry. Source code is available at https: //gi thub. com/prriiee nnctonn ips neu ra fdd.
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2 PHYSICS-CONSTRAIIE NEURAL FORM DISCOVERY
Our goal to generate mechanically efficient shapes for archi- tectural structures that approximate target geometries in real time while maintaining mechanical integrity. The challenge that, because mechanical efficiency is important in architectural structures, it is necessary to reason about designs from the point of view of force balance; but the resulting geometries are nontrivial function their mechanical behavior. Thus we seek to machine learning to efficiently invert this function to generate target designs without compromising their integrity.
P
[K(W)
2.1 COMPUTING EFFICIENT GEOMETRY
We focus on structures modeled as pin-jointed bar systems of N nodes connected by M bars (Figure 2). Each node experiences an external load vector (e.g., self-weight or wind load) and some nodes are constrained to fixed positions (e.g.. terrain and anchors). These are the structure's 's boundary conditions b E R-, After picking bar stiffnesses q E R-M, the goal of form-finding is to identify the positions of the nodes X (X1..... .,) N x3 such that there is no net residual force on the structure.
Figure 2: A bar system. In the callout, the stiffiess component [K(q)]ss ky is in- dicated at
connecting nodes with positions x; and x,. A load p. is ap- plied at xx. In this system, the nodes on the perimeter are fixed.
An efficient structure is whose loaded configuration is bending- and torsion-free, therefore reducing the material volume required to resist external loads. This configuration minimizes the structure's s strain energy by reducing the contribution of such components, letting a structure sustain loads mainly under tensile and compressive axial forces. We can satisfy this property if the shape of a structure modeled
bar system is in equilibrium. To arrive equilibrium, the residual force vector must be zero. The function r, rx;;; q) quantifies the difference
E IR3 at a free node with position between the load p. E R3 applied to node and the internal forces from the bars connected for given positions, stiffness values, and boundary conditions:
r, (X:q) JEN())
Σ
[(g)),s - xx)) (x.)
(1)
where (2) are the node neighbors, and K(q) E NNN is the stiffness matrix as function of The restriction that there is no residual force can be framed as constraint in which all of the r (X:q) q) = at the free nodes. Although this constraint can be solved with mechanical simulators like form-find form-finding methods (Adriaenssens et al., 2014), the resulting map from stiffness to geometry, which we denote X(q). is implicit and nonlinear, and difficult to reason about directly.
2.2 DIRECT OPTIMIZATION FOR TARGET SHAPES
Even though form-finding methods have the appealing property of promoting mechanical efficiency, they do not allow a designer to directly target particular geometries. Moreover, not all arbitrary geometries are even compatible with mechanical efficiency. If a designer has a target shape X, they wish to solve the following optimization problem with respect to q to approximate X with X:
NN
9"' = arg min Cahupe (q) QERM
where
Lshupe (q) =
[X())) - [
(2)
i=1
and p > We call Lehape the shape loss. This objective, which we refer to as direct optimization, identifies bar stiffness values that generate shapes close to the designer's intent in an E, sense while maintaining net zero force balance. Note that the optimization setup does not contain any information about the set of physically valid forms and it is simply driven by the minimization of the pointwise difference between shapes. Conventionally, this nonlinear optimization problem is solved numerically in the inner loop of design process, but that is slow and computationally costly (Figure 3).
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Target
Optiznisnaion
Solution
X
X(())
221.4
121.7
41.6
0.2
1/2
Figure 3: Shape matching. To generate a mechanically efficient shape X(q) that approximates an arbitrary target X, traditional methods like direct optimization find bar stiffnesses that minimize the shape loss Lahape. but only after several iterations L. Our model amortizes this computationally taxing process during inference while guaranteeing physics, enabling real-time and sound design.
2.3 AMORTIZED SHAPPE MATCHINNG
Rather than performing many costly optimizations within a design loop, we use machine learning model amortize the solution over family of target shapes X'. neural network takes as input the designer's intent X, and outputs the stiffness values q such that X(q) R X. Our model architecture resembles an autoencoder (Figure 1). First, a neural encoder & maps the target shape X into q. Then, mechanical simulator decodes the associated shape ((g), subject to the boundary conditions b. The key property of this construction is that the resulting shape is mechanically sound even if the neural network is inaccurate; the failure mode is not lack of mechanical integrity, but shape that does not match the target very well.
Neural encoder The encoder is a neural network with learnable parameters é that ingests the targets and projects them into stiffness space, &g R + RM For simplicity, we cast our problem as a point-wise matching task and use a multilayer perceptron (MLP) as the encoder, although we are not restricted to that. Regardless of the neural network specification, the output representation must be strictly positive. This necessary for compatibility with our mechanical simulator to avoid null stiffness values that bear limited physical meaning in our representation of an architectural structure. We satisfy this requirement by applying a strictly positive nonlinearity to the last layer of the encoder.
One of the advantages of our mechanical simulator is that it enables us to prescribe tensile or compressive bar forces a priori. As a result, rather than making these force directions a learnable feature, we build this bias into the encoder architecture by scaling the strictly positive embedding of the last layer by a force direction vector s E RR The scaling factors s E {-1, 1} indicate the direction of the internal axial force of every bar: negative factor prescribes compressive force, and a positive factor, a tensile force. Our encoder thus calculates the bar stiffness values q as:
q= ④ (ah + +)
where a is the strictly positive nonlinearity, h is the encoder's last layer embedding, - 2 0 is fixed scalar shift that specifies a minimum stiffness value for the entries of q (akin to a box constraint in numerical optimization), and @ indicates the element-wise product. Setting a lower bound on the stiffness values with J guides the learning process towards particular solutions since the map from q to X is not unique (Van Mele et al., 2012), and provides numerical stability when amortizing over structures with complex force distributions (Section 4.2).
Mechanical decoder represents the inputs of a mechanical simulator. To fulfill equilibrium in a pin-jointed bar system, the relationship between the stiffness values q. the shape X, and the loads P E RN must satisfy K(q) X - PX) = O. Solving this equation for bar system using finite element-based simulators (Xue et al., 2023; Wu, 2023) is possible, but requires second-order methods. Controlling the force signs adds numerical complexity, but is a desirable property to design structures built from tailored materials that strong only in tension compression (e.g., masonry blocks steel cables). Here, we utilize the force density method as our simulator (Pastrana et al., 2023b). Appendix A offers an extended description, but high level, the simulator is form-finding method that linearizes the equilibrium constraint by assuming independence between stiffness, geometry, and loads. This reduces the computation of X(q) RM + N with target force signs to linear solve:
The decoder gives the latent space of our model physical meaning since it
X(q) K(q) -pp
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10
Oute PINN NX
PINN
Phesseer Ejtsses
Shape less Caas
1gk
Stuge
PINN
NN
10-4
200
2.00
3000
1000
2030
6000
1000
2000
1063
122
415
Steges
Staps
Interpolation Eattoo, 6
(a) Train curves for Lshape and Ephyses
(b) Eshppe on interpolated data
Figure 4: Loss curves of the shell design task. (a) Our model learns meaningful representation that minimizes the shape loss Shhpp while fully satisfying the mechanics of compression-one shells, as Lphysies zero within numerical precision throughout training. (b) Shape loss of our model and the
PINN baseline on test data interpolated between doumly-symmettric (ō = 0) and asymmetric (8 = 1) geometries. Our model's ss accuracy decays at lower than the PINN's accuracy.
Training To amortize the shape-matching problem, we look for model parameters o² that minimize the expected value the shape loss:
Φ- arg min
Ex
ΣD d=1
[X]c
We train our model via first-order stochastic gradient descent, averaging the loss values over batches of size B each training step. Appendix provides training specifications. We generate training data by sampling batches target shapes X from a task task-specific family of shapes X parametrized by probability distribution (Section 4). Our model can be trained end-to-end because the encoder and decoder are both implemented differentiable programming environment (Bradbury et al., 2018). As result, reverse-mode automatic differentiation seamlessly backpropagate the physics-based gradients that tune the neural network parameters.
EVALUATION
We evaluate model performance by measuring the shape loss Lahape and the inference wall time test dataset of target shapes. We compare the performance of our model to that of three other baselines: a fully neural model (NN), a fully neural model augmented with physics-informar loss (PINN), and direct optimization.
The fully neural approaches replace the differentiable simulator in our model with a learnable decoder mirroring the encoder's 's architecture, with the addition of the boundary conditions inputs. The first baseline is trained to minimize the shape loss Lshape alone, highlighting that an information bottleneck is insufficient generate physically plausible designs. The second baseline extends the first baseline by adding an explicit physics loss term:
Cphsses Ex [|R(X(E,(X)))I
(6)
The physics loss measures the Frobenius norm of the residual forces R E N, at the N, free nodes of structure (see Appendix A). For shape to be physically valid in our setup, Ephysic must be zero within numerical precision (e.g.. 1 10 101)), We reason that the additional term should provide training signal to the encoder and decoder of the PINN so that they learn to solve the shape-matching task and the physics concurrently. The third baseline takes advantage of the differentiable mechanical simulator and optimizes q directly to minimize the shape loss via deterministic gradient- based optimization per-shape basis.
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(a) NN
(b) PINN
(c) Optimization
(d) Ours
&
x
9ms
Lauqo 1.3
Carp Eppsiss 0.7
Lau 1.1
Lamp 3.5
Cppriis 159.2
Lpysis 0.0
Cptesic 0.0
q
Cauge
Eshage
Cáuee ppayis 0.0
Estage Cyperics 0.0
Cyposis 44.1
Cytesics 0.5
Figure 5: Shape matching for shell design. While the NN and PINN models approximate the targets, they cannot suppress the residual forces (pink arrows). The stiffnesses and the shapes predicted by model are similar direct optimization's, ss indicating our model learns good task representation.
Table I: Model evaluation on the shell design task. We report the mean loss values and standard deviation test dataset of 100 target shapes, in addition to the inference wall time.
NN
PINN
Optimization
Ours
Estage
1.5 + 0.4
3.1 + 1.2
0.8 + 1.2
3.0 + 2.0
Ephysics |
104.3 + 48.6
0.6 + 0.3
0.0 + 0.0
0.0 1 0.0
Time [ms]]
0.3 + 0.1
0.3 + 0.1
5570.4 + 1688.6
0.6 + 0.1
4
EXPERIMENTS
We assess our model's ability to amortize shape approximation tasks for masonry shells and cable-net towers. These tasks represent broad class of design problems in structural engineering.
4.1 MASONRY SHELLS
Our first experiment determines suitable stiffness values for unreinforced masonry shells. Masonry shells sustain external loads with span-to-thickness ratios low 1:50 despite being built from materials that are strong in compression and weak in other loading conditions (Block et al., 2017). Shapes that maximize internal compressive axial forces enable this efficient behavior.
An expressive family of shells can be constructed with surfaces parameterized by a Bezier patch. The shape of the patch is in turn described by the positions of grid with C control points (see Appendix C.I). For this task, we restrict the space of target shapes to square grid of width w = 10 and C' = 16 control points. In particular, we consider doubly symmetric shapes with N 100 and M 180. We employ limit state analysis to model masonry shells as pin-jointed bar systems (Maia Avelino et al., 2021), and apply a constant area load of 0.5 per unit area, representing the self-weight of the shell. The nodes on the perimeter of the structures are fixed.
To solve this task, we are interested in finding bar stiffness values that yield shapes that fit the target geometries, and whose internal forces are compressive (q < 0). We satisfy the compression-only requirement in the last layer of our encoder by setting s = -1 We use 7 = 0 and = 1. Figure 4a illustrates the stochastic loss curves during training for model and the two neural baselines (NN and PINN). The neural baselines achieve a low shape loss but are unable to converge w.r.t. the physics loss, unlike our model where this requirement is satisfied by construction.

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Saaps 16.6
Interior view
Collapse sequence
(a) Model prediction
(b) Physical prototype
Figure 6: (a) Our model accurately predicts asymmetric shapes despite being trained exclusively on doubly symmetric geometries. (b) We build a predicted shape as tabletop prototype of a masonry shell. The bricks stand in equilibrium due to the appropriate shape; they are not mechanically attached as shown by the snapshots of the collapse sequence caused by an external perturbation.
The trends observed during training are consistent during inference (Table 1). Direct optimization achieves the lowest shape loss, but all the neural amortization models are faster than optimization. While the NN and the PINN generate close matches, their predictions are mechanically unfeasible because the residual forces do not vanish. The magnitude of the physics loss indicates that the structure is missing balancing forces to achieve equilibrium. For example, if we prescribe values for masonry shell with a surface area of 10m², thickness of 0.025 m. and material density of 2000 kg/m², S then residual force of 1 kN, representative of unit value of Cphysics in Table 1, would destabilize the structure with an acceleration of 2 m/s?. Consequently, the NN and the PINN predictions are unstable, and constructing masonry shells guided by them can lead to cracks, joint separations, and ultimately to collapse. In contrast, from the outset. Although the loss of direct optimization is lower, our model still generates accurate predictions up four orders of magnitude faster, and provides similar speedup for predictions that incur an equivalent shape loss (see Appendix D. Figure 5 shows two representative targets alongside the shapes, stiffness values, and residuals predicted by our model and the baselines.
model and direct optimization fulfill the physics constraint
Next, we investigate the out-of distrii-ittiitio generalization of our model and the PINN, an important property for building robust surrogates for physical design. To this end, we first generate a test dataset of 100 asymmetric Bezier surfaces and create target geometries by linearly interpolating between the existing doubly symmetric dataset and the new asymmetric dataset (Appendix C). We then evaluate the shape loss at increasing interpolation factors 8, where ā = O and ô = correspond to the doubly symmetric and asymmetric datasets, respectively. Note that both models have been trained solely on targets with double symmetry. A similar analysis for Lphysies is provided in Appendix D.2. Our model exhibits superior dutto--distribution performance. As shown in Figure 4b, the shape loss of model increases at half the rate of the PINN and with a lower variance. At 8 1, our model's shape loss is 2.5 times lower than that of the PINN. This performance gap is further illustrated in Figure 13 in Appendix D.2, where the PINN's prediction deviates from the asymmetric target by an order of magnitude more than in the doubly symmetric case, with physics loss two orders of magnitude higher. In contrast, our model's shape loss increases by only one order of magnitude, while the physics loss remains at zero, evidencing that physics-in-the-loop models like ours offer enhanced generalization compared to physics-ifformed approaches.
To demonstrate the practical applicability of our model, Figure 14 in Appendix D.3 shows its integration into a 3D modeling environment, where it assists designer in exploring various shell geometries. We further validate our model's generalization and transfer to physical applications by fabricating tabletop masonry shell based on one of our model's predictions (Figure 6a). The asymmetric shell features a thickness-to-span ratio of 1:50 along its longest span. After tessellating the predicted shape, we manufacture and assemble the resulting bricks. The prototype remains stable and resists its weight without glue or fasteners, proving that the structure primarily resists gravity through internal compressive forces as required for masonry shells (Figure 6b). Moreover, in Appendix D.4, solve this task at various discretizations and demonstrate that our model remains effective for real-time design at finer resolutions, all while maintaining prediction accuracy. These experiments confirm the feasibility of our model under real-world constraints.
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(a) Task
(b) PINN
(c) Optimization
(d) Ours
h
D
f
0:2
Lauge 6.9
Laxe 0.4 Ephyic
Latage 1.4
Cptycce
Eptyii
Figure 7: Predictions on cable-net structures. (a) Schematic depicting the design space. (b) (d) Reconstructions of target shapes, showing internal tensile (red) and compressive (blue) forces f. The PINN fails to match the target ellipse or ensure a net zero force balance, whereas our method closely approximates the target shape, similar the solution obtained through direct optimization.
4.2 CABLE-NET TOWERS
Guided by the success of our initial experiment, we turn to more complex problem: the design of cable-net towers. Such structures are lightweight, with low structural mass-to-volume ratios, and often materialized cooling observation towers (Schober et al., 2014). To explore the design space of these towers, this task focuses on approximating the shape and orientation of horizontal compression ring connected two vertically spanning, tensile cable-nets.
Figure 7a gives an overview of the task setup. Every tower comprises D = 21 rings with 16 points each, spaced at equal h D intervals over height of h 10. The discretization of the strueture is N = 335 and M = 656. We parametrize the geometry of the bottom, middle, and top rings with ellipses of radii 0x and agr,, with h/5, and in-plane rotation angle (Appendix F. The nodes on the top and bottom rings are fixed. Besides matching the shape of the compression ring, we look for geometries where the tension rings are planar. The shape loss incorporates these requirements, measuring the squared b norm (i.e., p 2) between predictions and targets. We set the entries of 10-1 and 1 impose the force sign of the compression ring and the tension cables, respectively.
The mechanical behavior of cable-net towers poses model- ing challenges because their structural members are either in tension compression. This interplay results in ill- conditioned problem, where interactions between elements of opposing force signs can lead singular systems and instabilities near zero force levels (Cai et al., 2018). While ill-conditioning can be managed in the forward pass, it can hinder learning in the backward pass by producing poorly scaled gradients (Figure 8). To mitigate these numerical instabilities, we clip the global gradient norm to 0.01 and shift the outputs of our last layer, enforcing lower bound of = in the stiffness space. Additionally, we incorporate Lreg into the training loss encourage our model to learn balanced stiffness distributions:
10
Clipping
Cahape
10
No clipping
Shape loss,
101
10
10
Steps
Figure 8: Training loss curve the tow- ers task, with and without clipping.
regularization
Lreg Var (4pos) + Var (Aneg)
This regularizer measures the variance, Var(q), of the bar stiffness values, for tensile Opos and compressive qneg components, over all the B samples in a batch. We scale Creg by constant factor We employ À = 10 to train our model and the baselines.
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Table 2: Model evaluation on the cable-net towers task. We report mean loss values and the standard deviation on a test dataset of 100 target shapes, in addition to the inference wall time.
NN
PINN
Optimization Expert
Ours
Randomized
with Ours
Latop
7.5 + 3.4
3.4
16.2 28.2
0.3 0.3
0.2 + 0.3
0.6
Lphysics
45.8 + 0.4
1.2 + 0.1
0.0 + 0.0
0.0 + 0.0
0.0 + 0.0
0.0 + 0.0
Time [ms] 4
0.5 + 0.1
0.5 1 0.1
1690.4 + 1169.9
1685.8 + 663.9
1460.5 + 823.4
4.6 +: 0.1
Figures 7b-7d show an example of the predicted cable-net tower shapes. Appendix E.2, we show that our model predictions cover the task space satisfactorily, generating accurate and mechanically valid cable-net shapes whose compression ring radii change size between r/2 and 3r 3r/2, and whose in-plane rotation vary within angle range of /6. Table 2 demonstrates that our model generates shapes that match the targets with a tighter fit than the NN and the PINN. These purely data-driven physies simulator, but they are unable to
baselines make faster predictions because they do not learn the physics of this task because Cphysies is nonzero, as we also identified in the shells task.
Furthermore, compare the two approaches that guarantee physics: ours and direct optimization. The convergence of direct optimization depends on selecting appropriate initial stiffness values In Figure 9, we therefore analyze the convergence rate of direct optimization with four different initializations. Random initial guesses of q converge to poor local optima as the shape loss is the highest among our experiments, underscoring the relevance of careful initialization optimize structures with complex mechanical behavior. If q is handpicked by a human expert, optimization is more accurate than our model and the other baselines. Optimization with expert initialization achieves a shape loss equivalent to that of our model in fifth of the convergence wall time reported in Table 2. Nevertheless, our trained model matches optimization in only one inference step that is three orders of magnitude faster and free of potentially expensive human intervention. These attributes make our model a more suitable approach to power automatic and interactive design tools.
Lastly, we investigate the effect of using the values of q predicted by our model and the PINN as inputs to direct optimization to refine the cable-net tower designs. This combination results in the most ac- curate matches, converging faster and consistently achieving a lower shape loss than direct optimiza- tion with expert initialized parameters (Figure 9).
10
Randlm
PINN
Shape loess Lahape
Expert
Ouz
10
The PINN initialization converges more slowly and only matches the performance of our initialization towards the end of the convergence curve. In both cases, a neural model provides better initialization for optimization than the human expert, highlighting the potential of utilizing neural networks and stan- dard optimization techniques in tandem to enhance mechanical design performance.
10"
10
350
500
1000
1500
Time
[ms]
Figure 9: Convergence curves of direct opti- mization with four distinct initializations.
RELATED WORKS
Differentiable mechanical simulators Machine learning and automatic differentiation have suc- cessfully been applied to obtain derivatives of complex forward mechanical models by either learning a differentiable surrogate with neural network, or by implementing the analytical physics model in differentiable programming environment. For neural surrogates, Xue et al. (2020) trained an autoencoder to simulate planar metamaterials, Zheng et al. (2020) developed a model to solve graphic statics problems on shells, and the work of Mai et al. (2024) focused on learning simulate cable-nets under various boundary conditions. Differentiable mechanical simulators include the finite element method for solids (Xue al., 2023) and isogeometric analysis of membranes (Oberbichler et al.,

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Published as a conference paper at ICLR 2025
2021). Specifically for form- form-finding, differentiable simulators exist for matrix structural analysis (Wu, 2023; Lee et al., 2024), and for combinatorial equilibrium modeling (Pastrana et al., 2023a). Our work leverages one such simulator and integrates it with a neural network to amortize inverse problems.
Amortization models for mechanical design Neural surrogates that amortize inverse problems have garnered attention for their ability to approximate nonlinear relationships in mechanical systems between target properties and input parameters. At the centimeter scale, Bastek et al. (2022) addressed the inversion of the urructure-properyy map in truss metamaterials, enabling the discovery of optimal configurations. Sun et al. (2021) synthesized path trajectories that compensate for fiber deformation during additive manufacturing. Oktay et al. (2023) utilized amortized models to generate actuation policies to deform cellular metamaterials several target configurations. Focusing architectural structures at the meter scale, Hoyer et al. (2019) proposed neural basis for computing material distributions that minimize compliance for buildings in 2D, while Chang and Cheng (2020) amortized the design of the cross sections of the beams and columns of buildings in 3D for multiple load cases. Favilli et al. (2024) applied geometric deep learning and a differentiable simulator to gridshell structures for low strain energy shapes. Unlike their work, which trained a neural network for single problem, we amortize over multiple inverse problems. Tam (2023) has amortized shape-matching problems with wonn-findin,, similar to our work, but their approach relies entirely on neural networks, which must learn both the physics and the matching task simultaneously.
CONCLUSION
We present a physics-in-the-lopp surrogate model that expedites the solution of shape-matching problems to design mechanically efficient geometry for architectural structures. By embedding prior physics knowledge into a neural network and training it end-to-end, our model learns representations that solve family of inverse problems with precision while enforcing mechanical constraints by construction, which is where purely data-driven approaches fall short. While we expect that PINNs could eventually push the physics loss closer to zero in the limit of hyperparameter tuning and network scaling, the physics guarantees and superior generalization performance of our model make it more robust approach to support design. Our model predicts mechanically sound geometries in milliseconds, with accuracy comparable to gradient-base based optimization. The speed and reliability of model enable real-time design exploration of long-span structures modeled as pin-jointed bar systems, such as masonry shells and cable-net towers.
6.1 LIMITATIONS AND FUTURE WORK
Although not cure-all for mechanical design problems, our work evidences that domain expertise in structural engineering and machine learning is necessary to address numerical pathologies caused by incorporating physics into neural networks (Wang et al., 2021; 2022; Metz et al., 2022). In our case, the simulator can yield an ill-conditioned system that affects training in the presence of structures with complex force distributions. Overcoming these instabilities requires knowledge of the physics of the problem being modeled and the application of gradient stabilization techniques common in machine learning, such as gradient clipping.
Like past methods this intersection, our model requires devising task-specific parameter spaces for learning (Allen et al., 2022). Consequently, a model trained for one task may not generalize to other tasks. Another limitation is that our model is currently restricted to one topology and has to be retrained when the discretization of a structure changes. In the future, we expect that applying methods such as graph networks (Battaglia al., 2018) to our encoder will enable us move beyond a single bar connectivity. We additionally note that the choice of bar stiffnesses for a given structure not unique and it potentially appealing to present to the designer a diversity of possible solutions by reformulating our model in a variational setting (Kingma and Welling, 2014; Salamanca et al., 2023). That is another exciting avenue for future research.
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ACKNOWLEDGMENTS
The authors thank Deniz Oktay for the early discussions on this project. This work has been supported by the U.S. National Science Foundation under grant OAC-2118201.
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A MECHANICAL SIMULATOR
We employ the force density method (FDM) (Schek, 1974) as our mechanical simulator. The FDM is form-finding method that computes torsion- and bending-free shapes on pin-jointed bar systems with NN nodes and M bars, by mapping bar stiffness values q E RM to node positions X E RNxs, x3;
subject to boundary conditions E R4, The stiffness qm or force density of bar sets the ratio between its internal force fn and its length (m in the equilibrium configuration:
4m
(8)
Since lengths strictly positive, a negative Qm indicates an internal compression force in the bar, and positive Qm denotes a tension force. In the FDM, a portion of nodes of size N, is fixed (i.e., anchors), and another of size N,, is free to displace. The size of the partitions is arbitrary as long as N = N₂ + Nu. but generally Nu >> Ng. For the geometry of a bar system to be in equilibrium, the residual force vector r, at each free node must be zero. To satisfy this constraint, the FDM calculates the free node positions X. E RNN x3 by solving linear system:
(q) K(g)=' [P. K(q), X.]
where the submatrices K(q),, E NoNN, and K(q) E N.NN are created by the rows and the columns of the geometric stiffness matrix K(q) E R&XN that correspond to the free and the fixed nodes in the structure, respectively. We solve this equation system with a direct, dense linear solver.
The FDM assembles K(q) based the connectivity between bars and nodes (Vassart and Motro, 1999). The value of the ij-th entry in the stiffness matrix is defined as:
Emem() Im ifi=
K(g))u
2m
nodes and j are connected by bar m
(10)
otherwise
where M(i) denotes the bars connected to node i. The loads applied to the nodes P E Nx3 and the positions of the node anchors X, E N3 are the boundary conditions of the problem, so they inputs the simulator. To create the boundary conditions vector b, we concatenate P and X,, and then flatten the resulting matrix into a vector. Once we know X., we concatenate it with X, to obtain all the node positions X that characterize a bar structure in equilibrium.
B IMPLEMENTATION DETAILS
We implement our work in JAX (Bradbury et al., 2018) and Equinox (Kidger and Garcia, 2021). We use JAX FDM (Pastrana et al., 2023b) as our differentiable simulator and Optax (DeepMind et al., 2020) for derivatives processing. For our model and the neural baselines (NN and PINN), we train separate model instance per task to minimize the loss expectation over a batch of shapes of size B. For optimization, we directly minimize the loss for every shape in the batch, one shape at a time.
Training and inference for all models are executed on a CPU, on a Macbook Pro laptop with an M2 chip. This choice of hardware is motivated by the type of devices structural engineers have access to and also reveals the potential for enhanced model performance on more powerful hardware, like GPUs, in the future. We train the three models using Adam (Kingma and Ba, 2017) with default parameters, except for the learning rate, for 10,000 steps. The training hyperparameters per model are tuned to maximize predictive performance on the test dataset via a random search over three seeds.
We employ SLSQP (Kraft, 1994) and L-BFGS-B (7hu et al., 1997), two deterministic gradient-based optimizers, as the direct optimization baselines. We use their implementations al., 2022) with default settings. To make a fair comparison, we also perform all the direct optimiza- tion experiments on the same laptop and CPU. We run these gradient- based optimizers for at most 5,000 steps and with convergence tolerance of 16
JAXOPT (Blondel
In direct optimization, we impose box constraints on the bar stiffness values q (the optimization variables) to prescribe explicit lower bound on the force signs and magnitudes. This mimics the effect of the last layer of our encoder via T (see Section 2.3). We use the same values of 7 that we report in Section& 4 as the lower box constraints. Additionally, we set upper box constraint on each
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Table 3: Neural architectures. We show the composition of the two types of autoencoders we use in Section 4: fully neural model employed for the NN and PINN models, and ours with the differentiable simulator as an analytical decoder. this table, Linear(in size, out size) denotes linear layer with learnable weights and biases followed by nonlinearity; whereas Concat(in_size, out_size denotes the concatenation two vectors.
Task
Model
Encoder
Decoder
# Parameters +
Train time (s) 4
[| Linear(3N. H);
[Linear(M (M + L, H);
2x Linear( H, Hf):
2x Linear( H. HI);
535,412
85
NN, PINN
Linear M)]
Linear [I, 3NN)):
Concar(3NN, + JN,,,,,, 3NV)
Shells
(3N, H);
Ours
Lineer H, H);
Simulator (M L,3N)
254,900
140
Linear H, M)1
[Linear(3R, II);
[Linear(A) ( + II):
4x Linear( H, H); Linear H, M)]
4x Linear H, H); Lincar[ H, 3.NN
NN, PINN
1,245,216
136
C3ncat(3N, + 1,N,,,77
Towers
[L.near(3R, H);
Ours
Linen H, H); Linear( H, M)]
Simulator(M + L, 3N)
468,880
810
bar stiffiness q so that |g| S. 20, where denotes the absolute value. Note that we do not enforce an upper bound on in any of the neural models during training or inference.
Table 3 summarizes the configuration of the neural models studied herein. We utilize MLPs for the learnable components of the three models. The MLPs are sequence of linear layers followed by nonlinear activation function. The weights and biases per layer are initialized random from uniform distribution in the interval [-1/ 1 1/vP), where is equal to the layer's input size. We use an ELU (Clevert et al., 2016) as the activation function in every MLP layer except in the last layer. The MI. encoders utilize Softplus as the last nonlinearity to satisfy the strictly positive requirement of our simulator (Section 2.3), and the MLP decoders use the identity function. Our model is more compact than the neural baselines because it has fewer than half the number of trainable parameters. However, our model takes longer to train due to the computational cost of the mechanical simulator.
The MLP decoders in the NN and PINN baselines take input the stiffness values predicted by the encoder, in addition to a vector with the boundary conditions so that the learnable decoder sees as much of the information the mechanical simulator has access This is different from, for example, conventional PINN approaches where the boundary conditions should be learned from a loss function, in addition the problem physies (Raissi al., 201 9a; Haghighat et al., 2021); but akin to the approach proposed in other recent works where boundary conditions are treated as either hard optimization constraints or imposed as network inputs (Lu et al., 2021: Bastek and Kochmann, 2023; Mai et al., 2024). Here, we simplify the decoder's 's goal by imposing the boundary conditions by construction, like in the latter case. This why the boundaries of the shape targets are matched exactly in all of our examples. For simplicity, b only comprises the positions of the N, fixed nodes and the vertical component of the loads applied to the N nodes (i.e., the horizontal components are zero) per target shape. Therefore, b has size L 3N, + N for all the neural decoders.
The last layer the MLP decoders outputs a vector of size 3.N,, representing the position of the free nodes, X.,. This choice is motivated by the fact that the mechanical simulator solves the form- finding problem exclusively at the free nodes the bar system with Equation 9, and subsequently concatenates the known positions of the support nodes, X., assemble the full shape matrix X. We replicate this process in the MLP decoder, ensuring that it predicts the same amount of information as the simulator. We describe task-specific implementation details next.
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B.1 MASONRY SHELLS
For the shells task, we train MLPs with 2 hidden layers with H = 256 units each. The size of the input and output layers varies per Table 3. The batch size is B = 64. The learning rate is 3 x 10-5 for the fully neural baselines (NN and PINN), and 5 x 10-a for our model. Our model takes minutes and 20 seconds train. The fully neural alternatives, seconds. We utilize SLSQP (Kraft, 1994) as the direct optimization baseline, sampling random values of q from uniform distribution bounded by the box constraints as initial guesses.
contrast, take minute and 25
We train the PINN baseline to minimize a weighted combination of the shape loss Lahape (Equation 2) and the physics loss Cpaysics (Equation 6):
C = L-hupe + K.Cphysics
(11)
Different values of impact predictive performance on this bipartite task at inference time (i.e., si- multaneously matching target shapes and reducing the residual forces). To strengthen the PINN baseline, we tune in five increments, from K 10 to 102, and train separate instance of the model for each. We then pick the PINN that achieves the lowest unweighted loss on the test 10 during inference to evaluate performance). Table 4 shows results.
dataset (i.e., we set We find that employing ^ = 10a during training produces the best-performen PINN baseline, and we use this PINN for comparison with our model and the other baselines in Section 4.
Table 4: PINN evaluation the masonry shells task for five different coefficients applied to Cphysses during training. The table reports the unweighted loss values generated by the trained PINN model predictions at inference time the test dataset.
10
10
10"
10
10a
Cstage
1.9 + 0.4
1.5
0.3
1.4 + 0.3
3.1
+ 1.2
92.0
40.5
Cphysics 4
35.3 + 16.7
6.3
+
2.8
2.3
+ 1.0
0.6
+ 0.3
0.3
0.1
B.2 CABLE-NET TOWERS
In the cable-net towers task, we train MLPs with 4 hidden layers of size H = 256. In all the models, we set the batch size B 16, and reduce the size of the encoder input from 3N 3R, where N is the total number of nodes in the structure, and R = 48 corresponds to the number of nodes on the three rings (bottom, middle, and top) that parametrize this task. This choice reduces the total number of trainable parameters in the encoders and makes them less computationally intensive.
The shape approximation task for the towers is underspecified because it only prescribes target height values for the tension rings of the cable-nets, rather than specific target 3D coordinates. Therefore, we mask (i.e., multiply by zero) the predicted x and y coordinates of the nodes of the tension rings before evaluating the shape loss.
In this task, we train our model in two stages, reducing the optimizer's learning rate from one stage to the next. In the first stage, we optimize for 5,000 steps with learning rate of 1 x 10-3, In the second stage, we fine-tune our model for another 5,000 steps with lower learning rate of 1 x 10-4 The total training time is 13.5 minutes. We train the MLPs of the NN and the PINN with learning rate x over 10,000 steps. The training time of both models is minutes and 16 seconds. The global gradient clip value is 0.01 for all the neural models.
The loss function we utilize to train the PINN baseline contemplates the shape and the physics losses, in addition to the regularization term (Equation 7) scaled by À 10:
f.stuee + x.Chhssis + >hrreg
(12)
Like in the shells task, vary the value of the weight coefficient between 10-2 As shown in Table 5, training the PINN with λ = 10" yields the best performance on the test dataset. We use this PINN variant for comparison with our model and the other baselines.
2 and 10
We employ L- L-BFGS-B B (Zhu et al., 1997), a quasi-Nuc quasi-Newton optimizer, as the direct optimization baseline in this task. We initialize q for direct optimization with four different approaches: randomized,
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Table 5: PINN model evaluation on the cable-net tower task for different coefficients applied to L-physics during training. The table reports unweighted loss values generated by the trained PINN predictions on the test dataset inference time.
10" 2
10-1
10
10'
10a
Lshape L
42.8
+
17.8
9.5
1 5.5
7.4 + 3.4
4.2
+
2.6
5.1
3.5
Ephysics
1.1 + 0.1
2.1 + 0.0
1.2 +: 0.1
3.8
4
0.7
4.3
0.6
expert, with the trained PINN, and with our trained model. The randomized initialization samples q from a uniform distribution bounded by the box constraints. The expert initialization sets q Both initialization approaches set the sign of the bar forces as specified by the task: negative signs for compression and positive signs for tension.
C DATA GENERATION ON BEZIER SURFACES
Bezier patch B maps a matrix C RC33 of control points smooth surface in R B RCx3 (u, v), parameterized by local coordinates u, [0, 1J. This parametrization offers clear control over architectural design intent as it enables the exploration of a wide array of smooth geometries by simply changing the positions of a coarse control grid.
C.I DATA GENERATION FOR SHELLS
summary of the data generation process is given in Figure 10. To generate a family of shapes for the shells task in Section units wide centered on the origin. The grid contains C = 16 control points arranged in a 4 x 4 layout.
focus doubly symmetric shapes produced by square grid
10
We then follow three main steps. First, we vary the 3D coordinates of the control points с₁ to C3 quarter of the control grid. The position of C4 is static. This construction ensures double symmetry in the generated data. To vary the position of every movable control point, we first sample translation vector t at random from uniform distribution in the interval [tmia tmax): ): and then we add it to its reference position c,, such that = C, + We detail the reference positions and the translation intervals for each control point in Table 6.
Next, we mirror the four control points on the xz and the yz Cartesian planes to obtain the position of the remaining twelve control points on the grid (see the callout in Figure 10). The bounding box of the resulting design space has dimensions 2w x 2w x w. In the third and last step, we evaluate N
C
c
X
C2
Figure 10: The data generation process of doubly symmetric shapes for the shells task consists of, first, creating variations of the positions of the points e₁ to сз in comner of the control grid of Bezier surface, and, then, mirroring twice. We create target points &; by evaluating the Bezier at N different locations the surface.
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Table 6: Generation parameters to sample random variations of the 3D coordinates of the control points to Ca on a quarter of the control grid of Bezier surface.
Control point
Reference position, C,-
Lower bound, Emin
Upper bound, tmax
C1
u//,w/66 0]
0, 0, us /10] -w/2, 0,0 00, 00-w0/2.0
[0. 0, us]
C2
w//2w
[w/2, 0, w/2)
w/6,w/22
[0. 2/,
w/2, w
coordinate values u and e, equally spaced in VN x VN grid, to generate an equal number of points on the surface. The evaluated points represent the position X E Nx the nodes of the structure we employ targets train our model and the baselines. The coordinates of a point & E R on the Bezier surface are a function of a (u, v) coordinates pair:
8(u, u)
(u( (1) Cog
(13)
where 7 denotes Bernstein polynomial of degree 3, and Cog indicates the position of the Bezier's control points, indexed on the 4 x 4 grid.
C.2 INTERPOLATION OF BEZIER SURFACES
We utilize linear interpolation to blend between doubly symmetric and asymmetric Bezier surfaces. Since the targets are a function the control points matrix C, we directly interpolate between the control points matrix Csym of doubly symmetric surface and that of an asymmetric surface Casym to create one design:
Cinterp (1 5)C + &Casym
(14)
where 5 E [0. 1] is scalar interpolation factor and Cinter is the interpolated control points matrix. We vary all the control points in Casym by sampling random translation vectors from a uniform distribution with the same bounds in the doubly symmetric case (Table 6). We finally generate X from Cineup with Equation 13. Figure 13a provides example of the shapes resulting from interpolating two surfaces.
D ADDITIONAL STUDIES FOR SHELL DESIGN
D.1 COMPARISON WITH DIRECT OPTIMIZATION
Both our model and direct optimization satisfy the physical constraints a priori, but they generate predictions at drastically different speeds. Since optimization is iterative approach, we compare the evolution of the shape loss over time to that of our method, which predicts the bar stiffness values of structure in one step. Optimization reaches the best predictive performance among the baselines, but only after convergence, which takes over 5000 milliseconds, shown in Figure 11. Even though optimization matches the shape loss of our model in 2000 mil- liseconds (35% of the convergence wall time in Table 1), approach offers a substantial speedup as it generates designs of equivalent accuracy in millisecond.
Optinazaion Ours
Shape loes, Calage
20-
Time
[ns
Figure 11: Shape loss evolution of direct optimization for shell design.
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D.2 GENERALIZATION THE PHYSICS LOSS
In Section 4.1, discussed the generalization capacity of our model and a PINN the approaches that are aware of the inverse problem physics-by by observing the of change of the shape loss Lstape as we morph the targets between doubly symmetric (6 and asymmetric (6 1) datasets.
10
Plysics loss Cpiosios
:00
Dus PINK
We now quantify the impact of perturbing the test data dis- tribution on the models' capacity to preserve the problem's physics. Figure 12 depicts the evolution of the physics loss Cphysics logarithmic scale. The performance of the PINN deteriorates rapidly as the distribution shifts from the doubly symmetric to the asymmetric dataset, with the loss increas- ing by an order of magnitude.
10
6.0
0.2
0.4
0.5
0.5
1.0
Interpolation Jactor,
Figure 12: Physics loss changes as the shapes vary between å 0 and 8 = 1.
This finding underscores that the in-distribution performance w.r.t physics the PINN does not necessarily lead to out-of-distribution generalization. In contrast, model enforces the physics by design, so playsic remains constant and effectively zero because our mechanical decoder, regardless of the geometry of the targets. Figure 13 illustrates this trend with example.
8 0
1/3
8 2/3
8
(a) Target shapes
Caraxe 2.1 Eytpais
Eaasge 26.8 Lynyeex
Lap: 54.1
Came 81.2 Lpyya
Eptyccs 10.4
(b) PINN
Lagge Cytosica 0.0
Lagge E 0.0
Laxge Cyposii 0.0
Cenge 27.8 Cytorics 0.0
(c) Ours
Figure 13: Top view of predicted shapes for shell design. (a) The targets are interpolated with different interpolation factors 8. (b) The accuracy of the PINN decays quickly w.r.t. f.stupe and Lphysics increases. (c) Our model's shape accuracy deteriorates, but at lower rate, and the problem's physics are fulfilled by construction since Cphysies 0 within numerical precision.
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D.3 REAL-TIME DESIGN IN CAD SOFTWARE
We deploy our trained model in the 3D modeling software Rhino3D (Robert McNeel & Associates, 2024) as illustrated in Figure 14. Rhino3D supports traditional computer-aided design (CAD) work- flows and Grasshopper (Figure 14, top left), its visual programming extension, allows the creation of new software features via custom Python scripts. We load our model as a Grasshopper plugin via Python and test it to support the real-time exploration of shapes for shell structures.
We describe a design exploration session next. A designer draws Bezier surface in Rhino3D and imports it into Grasshopper. Then, as the designer moves the control points of the Bezier, the geometry updates automatically. Our model generates new compression-only shapes response to the changes (see the bar systems rendered in Figure 14). Note that on doubly symmetric shapes, but the adequate generalization it exhibits to asymmetric geometries makes it possible support the designer during their exploratory session.
detail in Section 4, we trained model
Figure 14: demonstration of our trained model providing predictions of compression-only shapes in real time. The first screenshot shows the environment running custom code. The screenshot in the second column and first row shows the initial input Bezier surface with control points. Every subsequent sereenshot shows the designer moving Bezier control points and model reacting to the designer's 's intent by approximating the shape with one that is compression-onle
D.4 TRAINING COST AMORTIZATION
We measure trade-off's between our model and direct optimization in terms of training versus test inference time in problems where N > 500 and M > 1000. N and M are the numbers of nodes and bars in structure, respectively. These experiments also reflect the overhead of the mechanical simulator during training, as increasing the problem size also enlarges K, the bottleneck in Equation 4, We keep the architecture of our MLP encoder, except for the input dimension of the first layer, and the output dimension in the last layer, matching the new values of N and M. The settings for training and direct optimization repeat the configurations described in Appendix B. The size of the test dataset is B 100, except in optimization for M 1012 where B = 10 due to hardware limitations.
Table demonstrates that our model remains a feasible approach to amortize inverse problems on denser structures. While training and inference times increase with problem size, the expected inference time at the finest discretization is just ten milliseconds, making still suitable for real-time design exploration. Remarkably, the shape loss Latape normalized by N changes marginally the problem size grows, revealing the ability of our model to generate adequate fits to the target shapes across several discretizations despite the minimal changes in the encoder architecture (Figure 15).
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Latape NN 1.7%
Casp / 0.6%
Lauqe/ / 2.8%
Liagpe IN 2.3%
Juis
Carge 5.4%
Caugs /NogsNNe 0.9%
Cauu N 2.5%
Covee 2.6%
Figure 15: Our model makes accurate predictions across multiple discretizations. Top row: N = 100. Bottom row: N 529. The predictions satisfy the problem physics and are made in milliseconds.
Table Comparison between our model and direct optimization for varying problem sizes in the shells task. We report the mean loss values and standard deviations normalized by N on test dataset, and the inference wall time. The column Opt lists the number of optimization problems required to match or exceed the training time of our model, which corresponds to the number of inference calls needed to justify our model's training cost.
N
M
Optimization
Ours
Lstuee /N [%]]
Time [s] 4
Laage/ / [%] 4
Test time [ms] 4
Train time [s]
# Opt
100
180
0.8 + 1.2
5.6 + 1.7
3.0 + 2.0
0.6 = 0.1
140
25
256
480
1.0 + 1.5
287.7 115.8
2.3
3.5
1928
529
1012
0.8 + 0.9
5483.1 + 1946.0
3.5 + 2.4
10.2 + 0.3
6405
2
The amortization cost of training our model decreases relative direct optimization the problem size goes up (Table 7). Compared to the reference case in Section 4.1, the time needed to match one shape with optimization grows by two orders of magnitude for two times more bars, and by three orders of magnitude for five times as many bars. As result, the number of shape matches needed to justify training our model lowered by a factor of three every time the number of bars doubles.
E SHAPE EXPLORATION FOR CABLE-NET TOWER DESIGN
E. DATA GENERATION
To generate data for the cable-net tower task described in Section 4.2, parametrize the geometry of the bottom, middle, and top rings with ellipses of radii air and agr, and rotation angle 3 on the plane. Once set the reference radius to h/5, create the training and testing datasets by sampling the scale factors a; E [1/2, 3/2) and the angles 3 E [- x 112) from a uniform probability distribution. See Figure 7a for graphical description.
E.2 SHAPP EXPLORATION
We carry out two experiments that modify the size and the rotation the middle compression ring in cable-net tower in order illustrate the range of our trained model's predictions. In both experiments, we fix aj = 02 = 3/4 at the top and bottom rings to form circular anchors. the first experiment, shown in Figure 16, we modify the value of on and 02 of the middle compressive ring from a1 1/2
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to a1 3/2, and from 02 1/2 to 02 3/2, in five steps. In the second experiment, shown in Figure 17, the compression ring is elliptical from the start (a1 = 1/2, a2 = 3/2) and we rotate on the horizontal plane by angle 3 between /12 and 7/12, also in five steps. The predictions in both experiments match the target geometry. Note that the distribution of the bar forces f changes per prediction maintain force equilibrium while matching the target shape of the compression ring.
1/2
3/4
a=1
5/1
3322
Figure 16: Our model predictions for the cable-net tower design task, where we gradually increase the radius scale factor of the target circle of the compression ring from 1/2 to 3/2.
8 3= -1112
2/24
3=0 0
B= n/24
8 n/12
Figure 17: Our model produces physically valid predictions for the design of cable-net tower with an elliptical compression ring, rotate the ring by angle 8 between - 12 and π 1/12.
24


Reviews:
[{"review_id": "h39cLoAY6R", "reviewer": "Reviewer_NVMt", "summary": "The paper introduces a physics-in-the-loop scheme for the design of architectural structures. The authors here use a neural network(MLP) to learn the mapping from the desired structural shape to intermediate mechanical properties (bar stiffnesses) and then use a physics model to generate a physically feasible shape approximating the target.  The authors then apply this method to the design of masonry shells and cable towers,  comparing it with two neural network baselines: one trained to produce physically feasible shapes and the other trained to produce feasible shapes while also ensuring mechanical stability. Results on both these case studies show the proposed approach outperforming both these baselines and being on par with numerical optimization.", "questions": "Overall, the paper is technically sound and proposes an interesting integration of ML and physics. However, there are some concerns as follows:\n\n1. The proposed hybrid approach requires running the physics model during inference, likely increasing training costs. While the authors report inference time, the training time is not provided. Given the limited representational capacity of the network, it must be retrained whenever the design is re-parameterized (a likely scenario during conceptualization). Reporting training times would enable a better assessment of the method’s real-world viability; if training significantly exceeds the duration of several optimizations, direct optimization might be preferable.\n\n2. Similarly, providing training time metrics for the other baselines would help clarify trade-offs and be useful in scenarios where slight accuracy losses are acceptable for performance gains.\n\n3. From Fig.9 , the optimization initialized with the proposed method converges quickly as compared to the other initializations. This leads to an interesting question of how the NN initialized optimizations would perform. This approach would have the benefit of outputting a guaranteed local minima while avoiding the additional physics overhead during training. Including these results would enhance the paper's contribution to the community.\n\n4. The case studies considered here have relatively low DOFs, and the physics relies on a linear FDM. The authors could discuss the viability of this approach for structures where linear FDM is inapplicable or for dynamic scenarios. In such cases—and even when FDM is applicable but the structure has much higher DOFs—would this approach remain feasible?\n\n5. Furthermore, the variation in MLP and PINN inference times is puzzling. Since the input sizes remain constant, one would expect the inference times for the fully trained networks to be similar. Could the authors comment on this discrepancy?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The idea of combining physics with ML for architectural design is promising, as it removes the need for generating costly training labels. Instead, the model can learn by integrating the physics model with inexpensive loss functions.\n2. The out-of-distribution performance is also interesting, as it potentially reduces the need for extensive variability in the training data.", "weaknesses": "1. The case studies provided are relatively simplistic and do not reflect real-world applications. The physics model used is also quite basic, limiting the method's applicability in practical scenarios.\n2. Additionally, as the authors acknowledge, even when trained, the proposed parameterization lacks flexibility and requires retraining whenever the design representation changes.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "0kqaoyld8r", "reviewer": "Reviewer_VHnA", "summary": "This manuscript presents an interesting application of AE-type of network architecture to facilitate design and optimization of mechanical structures, mostly in (building) architectural structures. Although the method presented is mostly ad-hoc, the manuscript is well written, the proposed method appears sound, and the potential application is clearly indicated.", "questions": "1. Does the term *fabrication requirements* imply that each bar in the shell examples has to follow additional constraints?\n2. Provide more discussion on the usage of $\\tau$ as pointed out in the previous section\n3. Revise the manuscript to discuss the decoder in the main body of the manuscript.\n\nMinor issues:\n1. Define (or discuss) *mechanical integrity* and *mechanical efficiency* earlier in the manuscript. What do they exactly mean?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper is nicely written, although there are a few small details which could make the paper more understandable for neurips attendees, who may not have prior knowledge on the mechanical structure presented.\n\nThe method presented appears sound. The architecture as shown in Fig 1, leveraging JAX, makes intuitive sense.", "weaknesses": "Concerns to the paper:\n\n1. Although it is not clearly explained, it appears that each trained model is closely related to the underlying architectural structure (masonry shells or cable-net tower). The question is whether the trained model is generalizable. Even for the shells, if $M$ (number of bars) and $N$ (the number of nodes) change, the model may have to be retrained. It's understandable that this does not reduce the practical value of the proposed method. \n\n2. The runtime measurements have such large variances that they are not meaningful. The author(s) should consider changing the evaluation platform (macOS has many background processes and is not a good platform for benchmarking), and deploy better runtime measurement libraries.\n\n3. In line 200 in Eq. (3), the fact that $\\tau$ can be used to specify minimal stiffness is very interesting. But there are no more follow-up discussion on this topic. \n\n4. Structure of the paper: the decoder is not discussed in the main body of the paper, mostly in the appendices. This need to be fixed. \n\nMinor issues:\nThe terms *mechanical integrity* and *mechanical efficiency*  should be explained early in the paper to make it easier for readers to appreciate the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "u4Vd8hFvgD", "reviewer": "Reviewer_jNkr", "summary": "This paper combines neural networks with a differentiable mechanics simulator to develop a model that accelerates the solution of shape approximation problems for architectural structures modeled as bar systems. The proposed approach achieves better accuracy and generalization than the neural network approach and gets comparable accuracy to direct optimization. The two design domains are masonry shells and cable-net towers.", "questions": "* Are the authors going to be releasing the code? As an application paper, it would be beneficial to the community if they were.\n* Does the chosen bar stiffness affect the loading on the design since a stiffer bar might lead to a heavier design? \n* Could you provide further clarification on how the stress direction is determined a-priori? \n* When training the neural network component, how is this loss accounting for permutation invariance with locations, or is there some implicit consistency in the ordering of the pin coordinates.\n* What mechanism is in place to prevent the model providing excessively large stiffness values for all the bars?\n* Could you explain why the masonry shell is modeled as a pin-jointed structure but is then physically built as a shell (and how the two structure types are related)?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "* The presentation of the approach is nicely structured, and the figures are clean and easy to understand. \n* As an application paper, the work demonstrates a nice implementation of combining a differentiable simulator (as the decoder) with a neural network (as the encoder). The authors also build a physical prototype (Figure 6) based on the output of the approach. It is highly unusual to take this extra step to build a physical prototype and should therefore be commended.\n* The results successfully highlight how using the simulator ensures that the final design results in a pin-jointed structure where the resolved forces at the internal pins are at equilibrium.", "weaknesses": "* The motivation for this approach is not entirely clear. The design pipeline appears to start with a $\\tilde{\\mathbf{X}}$ and then produces a set of stiffness values. These stiffness values, along with the boundary conditions are then fed to the mechanical simulator to result in the final design (presumably, with the neural network’s stiffness values):\n    * In practice, how does one come up with a design $\\tilde{\\mathbf{X}}$? \n    * How does the simulator only require $\\mathbf{p}$ and $\\mathbf{b}$ for shape prediction? It seems like the simulator is missing information to build a shell with $N$ pins. \n* The cost of amortization (i.e. the number of direct optimization runs to build the data set for training) does not seem to be mentioned as a limitation of the approach. Are there instances in the real world where amortization of the optimization routine would be needed? For example, with a training set, it seems like 100 x direct optimization is required to train the model. This overhead would need at least 100 runs of the proposed approach to make it well-motivated. In a real design pipeline, how often would it be expected to call such a function? It already runs in less than 2 seconds for cable-net towers and around 6 seconds for the masonry shell. Is this this wall-clock time a problem in practice? If so, the paper would benefit from this written explicitly. \n\nMinor\n* $p$ for norm and $\\mathbf{p}$ for loads are a bit confusing in terms of notation.\n* Suggestion: use test-time inference rather than inference for predictions, since inference can also refer to inference over parameters.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lcbLagHjrz", "reviewer": "Reviewer_H1FF", "summary": "The authors propose an AI-driven design algorithm for architectural structure that can generate mechanically feasible designs for bar systems according to a target geometry. Previous approaches such as pure neural networks cannot guarantee mechanical design requirements. Traditional direct optimization methods can satisfy mechanical criteria, however are inefficient. The authors propose a hybrid scheme, combining neural networks and mechanical simulators to achieve computational efficiency and satisfaction of mechanical constraints. The authors approach consists of a encoder-decoder neural network, where the decoder is replaced by a differentiable mechanical simulator. The encoder structure provides efficiency, while the decoder ensures the design satisfies mechanical constraints. Experimental results show that the authors method achieves satisfactory performance in terms of design and computational efficiency.", "questions": "See weakness section", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper presentation is very good, the motivation is clear, and the method is justified well. Experimental results are extensive. The authors also provide a real life sample designed from their algorithm, which the readers will find very interesting.", "weaknesses": "The authors could do some revision of the figure. In figure 1, it seems that 3 figures are simultaneously provided to the encoder network, although the text description indicates only one shape is provided. Please take care to avoid such confusion from readers. Also describe the dimension of the matrix K. In figure 3, it is hard to see how the x vectors are being updated. Zooming in and adding some texts will be better for understanding.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)