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

TL;DR

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 is introduced, enabling large-scale shape OUU for complex systems.

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 (17)

• 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 $25\%–75\%$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 $25\%–75\%$ 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 $25\%–75\%$ 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.

Theorems & Definitions (54)

• Remark 2.1: On notation
• Proposition 2.1
• proof
• Remark 2.2: On the distribution of the model parameters
• Theorem 2.1
• Proposition 2.2
• Remark 2.3: On \ref{['assumption:pde_properties', 'assumption:qoi_properties']}
• Theorem 3.1: Universal approximation via multi-input neural operators
• Theorem 3.2
• Proposition 3.1: POD reconstruction error bound