Shape Derivative-Informed Neural Operators with Application to Risk-Averse Shape Optimization

Xindi Gong; Dingcheng Luo; Thomas O'Leary-Roseberry; Ruanui Nicholson; Omar Ghattas

math.OCMar 3, 2026

Shape Derivative-Informed Neural Operators with Application to Risk-Averse Shape Optimization

Authors: Xindi Gong, Dingcheng Luo, Thomas O'Leary-Roseberry, Ruanui Nicholson, Omar Ghattas

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 ·

Abstract

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 $C^1$ 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
$\mathcal{U}$ with basis $\{\phi_{i}\}_{i}$	$\mathcal{U}\ni u\mapsto\mathbf{\Phi}_{r_{u}}^{*}u=\left(\langle u,\phi_{i}\rangle_{\mathcal{U}}\right)_{i=1}^{r_{u}}\in\mathbb{R}^{r_{u}}$	$\mathbb{R}^{r_{u}}\ni c\mapsto\mathbf{\Phi}_{r_{u}}c=\sum_{i=1}^{r_{u}}c_{i}\phi_{i}\in\mathcal{U}$
$\mathcal{M}$ with basis $\{\psi_{i}\}_{i}$	$\mathcal{M}\ni m\mapsto\mathbf{\Psi}_{r_{m}}^{*}m=\left(\langle m,\psi_{i}\rangle_{\mathcal{M}}\right)_{i=1}^{r_{m}}\in\mathbb{R}^{r_{m}}$	$\mathbb{R}^{r_{m}}\ni c\mapsto\mathbf{\Psi}_{r_{m}}c=\sum_{i=1}^{r_{m}}c_{i}\psi_{i}\in\mathcal{M}$
$\mathcal{Z}=\mathbb{R}^{d_{z}}$	Identity	Identity

Table 1: Encoding and decoding operations

Example	$d_{W,i}$	Initial LR	LR Scheduler, $\gamma=0.5$	Epochs	$r_{u}$	$r_{m}$	$d_{z}$
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 (

L^{2}(\Omega_{0})

) (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 $u_{\text{true}}$ . Top-right: predicted state $u_{\text{DINO}}$ using Shape-DINO, with 512 training samples. Bottom-left: difference: $u_{\text{DINO}}-u_{\text{true}}$ . 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

\alpha=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\%\textendash 75\%

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

\beta=0.95

and penalty weight

\alpha=0.001

. Solid lines show the mean relative error and shaded regions represent the

25\%\textendash 75\%

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

\beta=0.95

and penalty weight

\alpha=0.1

. Solid lines denote the mean relative error and shaded regions show the

25\%\textendash 75\%

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.

Panel of Figure 6 — Figure 6: Top: optimal upper boundary designs from CVaR-SAA optimization with quantile $\beta=0.95$ and penalty weight $\alpha=0.001$ . 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, $z^{\star}_{\text{ref}}$ 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 $\tau_{\text{target}}$ . Colored lines correspond to flux profiles computed with $u(m^{(i)},z^{\dagger}_{\text{DINO}})$ , $u(m^{(i)},z^{\dagger}_{\text{NO}})$ and $u(m^{(i)},z^{\star}_{\text{ref}})$ for $256$ independent parameter samples $m^{(i)}\sim\mu_{m}$ . Shape-DINO optimal design produces flux profiles that aligns more closely with the target flux $\sigma_{\text{target}}$ 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

\beta=0.95

and

\alpha=0.001

, using a smooth target flux profile different from Figure 6 . Both DINO and NO are trained with 512 samples, while

z^{\star}_{\text{ref}}

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

\beta=0.95

and

\alpha=0.001

, using a discontinuous target flux profile defined by

x_{2}=1

for

x_{1}\in[0,2]

and

x_{2}=2

for

x_{1}\in[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

\Omega_{\text{channel}}=(0,2)\times(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

\xi_{k,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.

Panel of Figure 12 — 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 $z^{\dagger}_{\text{DINO}}$ , 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

\beta=1

for entropic risk measure and penalty weight

\alpha=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 $\Gamma_{\text{left}}$ and outflow on $\Gamma_{\text{right}}$ . The remaining faces $\Gamma_{\text{free-slip}}=\Gamma_{\text{front}}\cup\Gamma_{\text{top}}\cup\Gamma_{\text{back}}$ act as impermeable walls, while no-slip conditions are imposed on $\Gamma_{\text{object}}\cup\Gamma_{\text{base}}$ .

Figure 15: Distribution of horizontal-force QoIs for the optimal designs

z^{\dagger}_{\mathrm{DINO}}

z^{\dagger}_{\mathrm{NO}}

, and

z^{\dagger}_{\mathrm{PDE}}

. 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

\bar{m}^{s}

. 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

z^{\dagger}_{\mathrm{DINO}}

z^{\dagger}_{\mathrm{NO}}

, and

z^{\dagger}_{\mathrm{PDE}}

. 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

\bar{m}^{s}

. 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.

Panel of Figure 17 — Figure 17: Optimal tower shapes $z^{\dagger}$ with streamlines evaluated under eastward flow at the mean parameter values $m^{\mathrm{East}}=\bar{m}^{\mathrm{East}}$ . 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	$x$	$\chi(X)$
Function	$u(x)$	$u(\chi(X))=(T_{\chi}^{-1}u)(X)$
Gradient (scalar function)	$\nabla_{x}u(x)$	$F_{\chi}^{-T}(X)\nabla_{X}u(\chi(X))$
Gradient (vector function)	$\nabla_{x}v(x)$	$\nabla_{X}v(\chi(X))F_{\chi}^{-1}(X)$
Divergence	$\nabla_{x}\cdot v(x)$	$\mathrm{tr}(\nabla_{X}v(\chi(X))F_{\chi}^{-1}(X))$
Volume element	$dx$	$\det F_{\chi}(X)dX$

Appendix

Table 4: Common coordinate transformations. Here

F_{\chi}:=\nabla_{X}\chi

is the deformation gradient tensor,

|\cdot|

is the Euclidean norm of the vector, and

(\cdot)^{T}

corresponds to the matrix transpose.

PDE Operator Learning

Partial differential equation (PDE) operator learning builds models that map inputs such as coefficients and initial conditions to solution fields.

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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

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