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Real-time design of architectural structures with differentiable mechanics and neural networks
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Couples an MLP encoder with a differentiable force density method simulator to amortize shape-matching for bar-system architectural structures (masonry shells, cable-net towers), giving mechanically valid shapes in milliseconds.
Problem
Inverse form-finding for architectural bar structures: quickly finding stiffness/force densities whose equilibrium shapes approximate target geometries, while guaranteeing mechanical integrity, which purely neural surrogates and PINNs fail to ensure.
Approach
An autoencoder-style model where the encoder predicts strictly signed bar force densities from target shapes and a differentiable force density method simulator decodes the equilibrium shape, so equilibrium holds by construction and the simulator is trained end-to-end through autodiff.
Key results
- On masonry shells, the proposed model has shape loss 3.0 ± 2.0 with physics loss 0.0 at 0.6 ms, versus PINN 3.1 with physics loss 0.6 and direct optimization 0.8 at about 5570 ms.
- On cable-net towers, the model achieves shape loss 0.6 ± 0.7 in 4.6 ms, versus NN 7.5, PINN 7.4 (both with nonzero physics loss), and randomized-init optimization 16.2.
- Using the model's predictions as initialization for direct optimization yields lower shape loss than expert-initialized optimization on cable-net towers.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace the learned decoder with a differentiable mechanics simulator | shape approximation for masonry shells · Masonry shells test set | physics loss: 104.3 ± 48.6 (NN) and 0.6 ± 0.3 (PINN) -> 0.0 ± 0.0 | Helped |
| replace the learned decoder with a differentiable mechanics simulator | shape approximation for masonry shells · Masonry shells test set | shape loss: 0.8 ± 1.2 (Optimization), 1.5 ± 0.4 (NN), 3.1 ± 1.2 (PINN) -> 3.0 ± 2.0 | Mixed |
| replace per-target direct optimization with amortized prediction | masonry shell design · Masonry shells test set | inference time [ms]: 5570.4 ± 1688.6 -> 0.6 ± 0.1 | Helped |
| use the physics-in-the-loop model instead of the PINN | out-of-distribution shell shape matching at δ=1 · Asymmetric masonry shells | shape loss: 2.5 times lower | Helped |
| replace the learned decoder with a differentiable mechanics simulator | shape approximation for cable-net towers · Cable-net towers test set | physics loss: 45.8 ± 0.4 (NN) and 1.2 ± 0.1 (PINN) -> 0.0 ± 0.0 | Helped |
| replace learned decoding with differentiable mechanics | shape approximation for cable-net towers · Cable-net towers test set | shape loss: 7.5 ± 3.4 (NN), 7.4 ± 3.4 (PINN), 0.3 ± 0.3 (Optimization expert) -> 0.6 ± 0.7 | Mixed |
| replace direct optimization with one-step amortized prediction | cable-net tower design · Cable-net towers test set | inference time [ms]: 1685.8 ± 663.9 -> 4.6 ± 0.1 | Helped |
| initialize direct optimization with Ours predictions rather than expert parameters (ablation) | cable-net tower design · Cable-net towers test set | shape loss: 0.3 ± 0.3 -> 0.2 ± 0.3 | Helped |
| initialize direct optimization with random stiffness values rather than expert parameters (ablation) | cable-net tower design · Cable-net towers test set | shape loss: 0.3 ± 0.3 -> 16.2 ± 28.2 | Hurt |
| train the PINN with physics-loss weight κ=10^-2 instead of κ=10^1 (ablation) | PINN for masonry shells · Masonry shells PINN test set | physics loss: 0.6 ± 0.3 -> 35.3 ± 16.7 | Hurt |
| train the PINN with physics-loss weight κ=10^2 instead of κ=10^1 (ablation) | PINN for masonry shells · Masonry shells PINN test set | shape loss: 3.1 ± 1.2 -> 92.0 ± 40.5 | Hurt |
| train the PINN with physics-loss weight κ=10^1 instead of κ=10^0 (ablation) | PINN for masonry shells · Masonry shells PINN test set | physics loss: 2.3 ± 1.0 -> 0.6 ± 0.3 | Helped |
Surprising findings
- Direct optimization with expert initialization on cable-net towers attains a lower shape loss (0.3) than the proposed model (0.6), so the model is not the most accurate option there.
- The PINN baseline generalizes worse out of distribution: at full asymmetry its shape loss is 2.5 times higher than the proposed model's.
Limitations
- Model is restricted to one topology and must be retrained when discretization changes.
- Ill-conditioned simulator with mixed force signs causes unstable gradients, requiring gradient clipping and lower-bounded stiffness.
- Choice of bar stiffnesses is not unique, and the model currently gives a single solution rather than a diversity of designs.
Experimental setup
Datasets: synthetic Bezier surface target shapes (doubly symmetric) asymmetric Bezier surfaces (test)
Benchmarks: masonry shell shape matching (100 test targets) cable-net tower shape matching (100 test targets)
Metrics: shape loss physics loss (residual force norm) inference wall time (ms)
Baselines: NN (fully neural autoencoder) PINN direct optimization (SLSQP) direct optimization (L-BFGS-B)
Builds on: force density method physics-informed neural networks JAX FDM
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