Organizations: Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas, USA · School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia · Centre for Data Science, Queensland University of Technology, Brisbane, Australia · Department of Mathematics, The Ohio State University, Columbus, Ohio, USA · Department of Engineering Science and Biomedical Engineering, University of Auckland, Auckland, New Zealand · Walker Department of Mechanical Engineering, The University of Texas at Austin, Austin, Texas, USA.
Shape optimization under uncertainty (OUU) is computationally intensive for classical PDE-based methods due to the high cost of repeated sampling-based risk evaluation across many uncertainty realizations and varying geometries, while standard neural surrogates often fail to provide accurate and efficient sensitivities for optimization. We introduce Shape-DINO, a derivative-informed neural operator framework for learning PDE solution operators on families of varying geometries, with a particular focus on accelerating PDE-constrained shape OUU. Shape-DINOs encode geometric variability through diffeomorphic mappings to a fixed reference domain and employ a derivative-informed operator learning objective that jointly learns the PDE solution and its Fréchet derivatives with respect to design variables and uncertain parameters, enabling accurate state predictions and reliable gradients for large-scale OUU. We establish a priori error bounds linking surrogate accuracy to optimization error and prove universal approximation results for multi-input reduced basis neural operators in suitable C1 norms. We demonstrate efficiency and scalability on three representative shape OUU problems, including boundary design for a Poisson equation and shape design governed by steady-state Navier-Stokes exterior flows in two and three dimensions. Across these examples, Shape-DINOs produce more reliable optimization results than operator surrogates trained without derivative information. In our examples, Shape-DINOs achieve 3-8 orders-of-magnitude speedups in state and gradient evaluations. Counting training data generation, Shape-DINOs reduce necessary PDE solves by 1-2 orders-of-magnitude compared to a strictly PDE-based approach for a single OUU problem. Moreover, Shape-DINO construction costs can be amortized across many objectives and risk measures, enabling large-scale shape OUU for complex systems.
Figures & tables
Encoding
Decoding
U with basis {ϕi}i
U∋u↦Φru∗u=(⟨u,ϕi⟩U)i=1ru∈Rru
Rru∋c↦Φruc=∑i=1ruciϕi∈U
M with basis {ψi}i
M∋m↦Ψrm∗m=(⟨m,ψi⟩M)i=1rm∈Rrm
Rrm∋c↦Ψrmc=∑i=1rmciψi∈M
Z=Rdz
Identity
Identity
Table 1: Encoding and decoding operations
Example
dW,i
Initial LR
LR Scheduler, γ=0.5
Epochs
ru
rm
dz
Poisson
[512,512,512,512]
0.0005
{800,1200,1800}
2000
400
256
11
2D NS
[512,1024,512]
0.0001
{500,750}
1000
256
50
162
3D NS
[512,512]
0.0001
{1000,1500}
2000
200
128
160
Table 2: Neural network architectures and training hyperparameters for all experiments. For each experiment, Shape-DINO and Shape-NO use identical settings, except for the loss function. Initial LR denotes the initial learning rate used by the Adam optimizer. LR Scheduler specifies the learning-rate stages, where the set of integers denotes the epoch milestones at which the learning rate is updated by a factor 0.5 .
Figure 1: Mean relative testing errors for full order state ( L2(Ω0) ) (left) and reduced order Jacobians (middle and right) of Poisson PDE neural operators trained with (Shape-DINO) and without derivative information (Shape-NO). Training with Jacobian information consistently yields lower errors in both the solution and its sensitivities, demonstrating improved accuracy and robustness of shape-DINO.
Figure 2: Example of state u plotted on deformed domain. Top-left: true PDE state utrue . Top-right: predicted state uDINO using Shape-DINO, with 512 training samples. Bottom-left: difference: uDINO−utrue . Bottom-right: parameter m used for the calculation. Visually, the prediction error is near-zero, confirming the accuracy of Shape-DINO.
Figure 3: Optimal cost values at optimal solutions (left) and relative error of the objective function (right) versus the number of state PDE solves required for the optimization, both with mean SAA and penalty weight α=0.001 . Results are shown for shape-DINO, shape-NO, and the PDE-based SAA. Solid lines denote the mean relative error, and shaded regions indicate the 25%\textendash75% quantile bands across multiple runs. Even when the objective function is easy to estimate (as in the mean objective), Shape-DINO converges rapidly to the reference solution with substantially fewer PDE solves than the PDE-based method, while Shape-NO exhibits larger errors and slower convergence.
Figure 4: Optimal cost values at optimal solutions (left) and relative error of the objective function (right) versus the number of state PDE solves required for the optimization, both with quantile β=0.95 and penalty weight α=0.001 . Solid lines show the mean relative error and shaded regions represent the 25%\textendash75% quantile bands. In this highly risk-sensitive setting, Shape-DINO achieves near-reference accuracy with orders of magnitude fewer PDE solves than the PDE-based SAA, whereas Shape-NO performs poorly for small and moderate training sizes, highlighting the importance of derivative information for accurate risk estimation.
Figure 5: Optimal cost values at optimal solutions (left) and relative error of the objective function (right) versus the number of state PDE solves required for the optimization, both with quantile β=0.95 and penalty weight α=0.1 . Solid lines denote the mean relative error and shaded regions show the 25%\textendash75% quantile bands. As the penalization term becomes dominant, the performance gap between Shape-DINO and the PDE-based method narrows, though Shape-DINO still provides the best trade-off between accuracy and offline computational cost, while Shape-NO remains significantly less reliable.
Figure 6: Top: optimal upper boundary designs from CVaR-SAA optimization with quantile β=0.95 and penalty weight α=0.001 . The three subplots show the optimal designs by Shape-DINO, Shape-NO, and PDE reference, respectively, together with their corresponding PDE state solutions u(m,z) on the deformed domain. Here, zref⋆ is obtained from a large-sampled PDE-based SAA with 16,284 samples. Bottom: sample flux profile at optimal designs. The black line denotes the target flux τtarget . Colored lines correspond to flux profiles computed with u(m(i),zDINO†) , u(m(i),zNO†) and u(m(i),zref⋆) for 256 independent parameter samples m(i)∼μm . Shape-DINO optimal design produces flux profiles that aligns more closely with the target flux σtarget compared to shape-NO and achieve performance very close to the PDE reference.
Figure 7: Sample flux profile at the optimal designs obtained from CVaR optimization with β=0.95 and α=0.001 , using a smooth target flux profile different from Figure 6 . Both DINO and NO are trained with 512 samples, while zref⋆ is obtained from a large-sampled PDE-based SAA with 16,284 samples. Despite using orders of magnitude fewer PDE solves in the offline stage, shape-DINO produces flux profiles that are nearly indistinguishable from those of the large-sample PDE reference, closely tracking the target flux across parameter realizations and exhibiting reduced variability, whereas shape-NO shows much larger deviations.
Figure 8: Sample flux profile at the optimal designs obtained from CVaR optimization, with β=0.95 and α=0.001 , using a discontinuous target flux profile defined by x2=1 for x1∈[0,2] and x2=2 for x1∈[2,4] . Both DINO and NO are trained with 512 samples, while the PDE reference is obtained from a large-sample PDE-based SAA. Despite the target flux lying outside of the distribution used during training, shape-DINO produces flux profiles that better match the target and remain stable across uncertainty, whereas shape-NO produces flux profiles that significantly deviate from the target.
Figure 9: Panel (left) shows the reference channel domain Ωchannel=(0,2)×(0,1) with the labeled boundaries. The obstacle is a circle with radius 0.1 centered at (0.5,0.5) . Grey dots are the Bernstein control points ξk,l . Panels (middle) and (right) show the corresponding deformed domains under eastward and northward displacement bases, respectively, with only domain boundaries displayed for clarity.
Figure 10: State (left) and Jacobians (middle and right) testing errors for eastward inflow case 2D Navier-Stokes neural operators with (Shape-DINO) and without (Shape-NO) Jacobian training. Training sample sizes are 512, 1,024, 2,048, 4,096 and test sample size is 1,024. Shape-DINO yields significantly lower generalization errors in both the outputs and the derivatives compared to Shape-NO.
Figure 11: State (left) and Jacobians (middle and right) testing errors for northward inflow case 2D Navier-Stokes neural operators with (Shape-DINO) and without (Shape-NO) Jacobian training. Training sample sizes are 512, 1024, 2048, 4096 and test sample size is 1024. Similar to the eastward inflow case, Shape-DINO again yields markedly lower errors in both the predicted state and its derivatives compared to Shape-NO.
Figure 12: Comparison of representative flow fields for the initial object geometry and the optimized geometry obtained using DINO trained with 1024 samples. Each row corresponds to one of the two inflow scenarios (eastward and northward flow), and each column shows the velocity fields before (left column) and after (right column) optimization. All flow fields are evaluated using high-fidelity PDE solves. The optimized geometry zDINO† , though not symmetric, significantly reduces recirculation behind the object and lowers the overall viscous dissipation rate compared to the initial shape.
Figure 13: Optimal cost values at optimal solutions (left) and relative error of the objective function (right) versus the number of state PDE solves required for the optimization, both with risk-aversion parameter β=1 for entropic risk measure and penalty weight α=0.001 . NN train sizes are 512, 1,024, 2,048, and 4,096, and all used 8,192 samples per iteration for ouu. PDE ouu per iteration with 16, 32, 64, 128, 512 samples. PDE reference ouu is done with 8,192 samples for iteration. The results are evaluated with the same 16,384 samples. Note that the counts reported above refer to the number of samples per flow scenario . The actual number of PDE solves is twice the reported number since we evaluate the QoI for both the eastward flow and northward flow scenarios.
Figure 14: Geometry of the domain and initial frustum-shaped tower together with streamlines of the velocity field obtained from the PDE Navier–Stokes solution. The case shown corresponds to the left inflow condition with inflow on Γleft and outflow on Γright . The remaining faces Γfree-slip=Γfront∪Γtop∪Γback act as impermeable walls, while no-slip conditions are imposed on Γobject∪Γbase .
Figure 15: Distribution of horizontal-force QoIs for the optimal designs zDINO† , zNO† , and zPDE† . All designs are evaluated with 200 Monte Carlo samples. The DINO and NO solutions are obtained with 1,024 samples per iteration, while the PDE solution is computed deterministically at mˉs . The Shape-DINO design yields a distribution that is both more concentrated and closer to zero, while shape-NO exhibits broader distributions with heavier tails, highlighting the benefit of incorporating derivative information for achieving robust designs under uncertainty.
Figure 16: Distribution of bending-moment QoIs for the optimal designs zDINO† , zNO† , and zPDE† . All designs are evaluated with 200 Monte Carlo samples. The DINO and NO solutions are obtained with 1,024 samples per iteration, while the PDE solution is computed deterministically at mˉs . The Shape-DINO distribution is much more tightly concentrated and shifted closer to zero, indicating reduced magnitude and variability of the bending-moment QoI compared to both the deterministic PDE and shape-NO designs.
Figure 17: Optimal tower shapes z† with streamlines evaluated under eastward flow at the mean parameter values mEast=mˉEast . Panels 17(a) , 17(c) , and 17(e) correspond to the horizontal force QoI, while panels 17(b) , 17(d) , and 17(f) show results for the bending moment QoI. The Shape-DINO designs are thinner and allow smoother flow passage compared to deterministic PDE, whereas shape-NO yields bulkier and sometimes twisted geometries that induce stronger flow deflection and larger downstream recirculation regions. This comparison provides a physical explanation for the reduced forces and bending moment achieved by Shape-DINO.
Execution time per sample (seconds)
State Solve
VJP / Adjoint
Problem
PDE
NN (1)
NN (1000)
PDE
NN (1)
NN (1000)
Poisson
1.06e − 1
4.12e − 4
5.50e − 7
8.84e − 2
1.25e − 5
1.35e − 6
2D Navier–Stokes
5.08e + 0
3.86e − 4
1.70e − 6
1.07e + 1
1.15e − 4
3.16e − 6
3D Navier–Stokes
7.18e + 2
3.85e − 4
5.46e − 6
1.62e + 2
9.68e − 4
1.07e − 5
Table 3: Time (in seconds) for solving the state PDE and computing vector–Jacobian products (VJPs) using neural operators and PDE-based solvers. PDE-based solver timings are averages over 100 independent random samples of (m,z) . For neural operators, NN (1) denotes inference performed on a single sample, while NN (1000) corresponds to batched inference over 1000 samples evaluated simultaneously, with the time normalized per sample. The reported times for NN (1) and NN (1000) are both averages over 100 independent trials. For neural operators, we report the time to compute the VJP by a backward pass, while for PDE-based method, we report the time taken to solve the adjoint PDE, which is the dominant cost of the VJP. Neural operators achieve orders-of-magnitude reductions in computational time compared to PDE-based solvers for both state and VJP evaluations, with particularly pronounced gains in the 2D and 3D Navier–Stokes cases. These gains are further amplified under batched inference (NN (1000)), where native GPU parallel evaluation leads to additional per-sample reductions in execution time.
Appendix figures & tables1 asset
Supplementary material from the paper’s appendix.
Appendix
Spatial coordinates x
Reference coordinates X
Coordinate
x
χ(X)
Function
u(x)
u(χ(X))=(Tχ−1u)(X)
Gradient (scalar function)
∇xu(x)
Fχ−T(X)∇Xu(χ(X))
Gradient (vector function)
∇xv(x)
∇Xv(χ(X))Fχ−1(X)
Divergence
∇x⋅v(x)
tr(∇Xv(χ(X))Fχ−1(X))
Volume element
dx
detFχ(X)dX
Appendix
Table 4: Common coordinate transformations. Here Fχ:=∇Xχ is the deformation gradient tensor, ∣⋅∣ is the Euclidean norm of the vector, and (⋅)T corresponds to the matrix transpose.
Engineering shape optimization faces challenges in both expert-dependent problem setup and surrogate-model reliability. In practical aerodynamic design, optimization settings such as editable regions, deformation ranges, and design-preservation constraints are typically specified manually by experienced engineers, while surrogate-based optimization may become unreliable for heterogeneous geometry databases and out-of-distribution designs. To address these challenges, we propose a knowledge-constrained shape-optimization framework that translates knowledge-based constraints and user intent into quantifiable parameters of DFFD-based deformation operators, enabling engineering-aware and controllable constrained optimization. We further develop a Mixture-of-Experts Neural Operator (MoE-NO) to improve drag prediction and trend consistency over heterogeneous aerodynamic datasets. Based on the MoE-NO encoder and Mahalanobis distance, an uncertainty-estimation strategy is introduced to detect out-of-distribution geometries and selectively trigger physics-solver feedback for local sample enrichment. Experiments on in-house MPV, SUV, and Sedan datasets show that MoE-NO achieves a test-set MAPE of 1.16% and a trend-prediction accuracy of 94.34%, outperforming the best baseline results of 1.52% and 90.34%, respectively. Vehicle shape-optimization experiments further yield CFD-validated drag coefficient reductions of approximately 4% to 10%.
Wenhao Fan, Yuanwei Bin, Jianghan Gu +4
School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China · 2Eastern Institute of Technology, Ningbo, Ningbo 315200, Zhejiang, China · 3TenFong Technology Co., Ltd., Shenzhen 518000, Guangdong, China +1
We prove error bounds for operator surrogates of solution operators for partial differential and boundary integral equations on families of domains which are diffeomorphic to one common reference (or latent) domain Dref. The pullback of the PDE to Dref via affine-parametric shape encoding produces a collection of holomorphic parametric PDEs on Dref. Sufficient conditions for (uniformly with respect to the parameter) well-posedness are given, implying existence, uniqueness and stability of parametric solution families on Dref. We illustrate the abstract hypotheses by reviewing recent holomorphy results for a suite of elliptic and parabolic PDEs. Quantified parametric holomorphy implies existence of finite-parametric, discrete approximations of the parametric solution families with convergence rates in terms of the number N of parameters. We obtain constructive proofs of existence of Neural and Spectral Operator surrogates for the shape-to-solution maps with error bounds and convergence rate guarantees uniform on the collection of admissible shapes. We admit principal-component shape encoders and frame decoders. Our results support in particular the (empirically reported) ability of neural operators to realize data-to-solution maps for elliptic and parabolic PDEs and BIEs that generalize across parametric families of shapes.
Geometry is central to PDE-governed systems, motivating shape optimization and inversion. Classical pipelines conduct costly forward simulation with geometry processing, requiring substantial expert effort. Neural surrogates accelerate forward analysis but do not close the loop because gradients from objectives to geometry are often unavailable. Existing differentiable methods either rely on restrictive parameterizations or unstable latent optimization driven by scalar objectives, limiting interpretability and part-wise control. To address these challenges, we propose Geometry-Aware Neural Optimizer (GANO), an end-to-end differentiable framework that unifies geometry representation, field-level prediction, and automated optimization/inversion in a single latent-space loop. GANO encodes shapes with an auto-decoder and stabilizes latent updates via a denoising mechanism, and a geometry-informed surrogate provides a reliable gradient pathway for geometry updates. Moreover, GANO supports part-wise control through null-space projection and uses remeshing-free projection to accelerate geometry processing. We further prove that denoising induces an implicit Jacobian regularization that reduces decoder sensitivity, yielding controlled deformations. Experiments on three benchmarks spanning 2D Helmholtz, 2D airfoil, and 3D vehicles show state-of-the-art accuracy and stable, controllable updates, achieving up to +55.9% lift-to-drag improvement for airfoils and ~7% drag reduction for vehicles.
Guoze Sun, Tianya Miao, Haoyang Huang +4
Gaoling School of Artificial Intelligence, Renmin University of China