PDE-Constrained Optimization Methods

• PDE-Constrained Optimization is a framework that integrates partial differential equations with optimization techniques to compute optimal controls, designs, or inputs for complex systems.
• It leverages methods such as adjoint-gradient schemes, interior-point strategies, and operator-splitting approaches to efficiently address high-dimensional and constrained problems.
• Applications span optimal flow and shape design, inverse modeling, mesh adaptivity, and emerging quantum and surrogate-based techniques for real-time, large-scale computations.

Partial differential equation (PDE)-constrained optimization is the task of finding optimal parameter values (controls, designs, or inputs) for systems whose state evolution is governed by PDEs, while directly enforcing the PDE constraints within the optimization. This structure arises ubiquitously in control, design, inverse problems, and data assimilation throughout applied mathematics, engineering, and scientific computing. Canonical examples span optimal flow and shape design, subsurface parameter identification, mesh adaptivity, structural optimization, and data-driven model calibration. Research in this field has generated a suite of mathematical theories, numerical algorithms, and advanced computational frameworks targeting well-posedness, scalability, and practical expressivity in high-dimensional settings.

1. Mathematical Foundations and General Formulation

The canonical PDE-constrained optimization problem is formulated as

$$\text{minimize } J(u, m) \quad \text{subject to } F(u, m) = 0,$$

supplemented by optional constraints $h(m) = 0$ and $g(m) \leq 0$ on the controls or design variables. Here $u$ denotes the PDE state (e.g., a function field such as velocity or temperature); $m \in \mathbb{R}^N$ aggregates control or design parameters; $J(u, m)$ is the scalar objective functional; and $F(u, m) = 0$ encodes (possibly nonlinear, time-dependent, or coupled) PDE constraints with appropriate boundary/initial conditions (Funke et al., 2013).

To enforce the PDE constraint in optimization, the Lagrangian is introduced: $$L(u, m, \lambda) = J(u, m) + \langle \lambda, F(u, m) \rangle,$$ with adjoint variable $\lambda$ dualizing the PDE constraint. First-order necessary conditions correspond to the stationarity (Karush–Kuhn–Tucker, or KKT, system): $$\begin{aligned} &\partial_u L = J_u + F_u^T \lambda = 0, \ &\partial_m L = J_m + F_m^T \lambda = 0, \ &\partial_\lambda L = F(u, m) = 0, \end{aligned}$$ where $F_u$, $F_m$ denote derivatives of $F$ with respect to $u$ and $m$ (Funke et al., 2013).

The adjoint method arises by solving, for each $m$,

$F_u(u, m)^T \lambda = -J_u(u, m),$

to compute the reduced gradient

$$\nabla_m \hat{J}(m) = J_m(u(m), m) + F_m(u(m), m)^T \lambda,$$

with $u(m)$ implicitly defined by $F(u, m) = 0$.

2. Computational Frameworks and Algorithmic Implementations

A variety of algorithmic strategies have been established:

• Reduced formulation (adjoint-gradient methods): Each optimization step solves the PDE "forward" for $u(m)$, then an adjoint PDE (backward) for $\lambda$, computes the gradient, and applies an external optimizer (e.g., quasi-Newton, L-BFGS, SLSQP) in the control space. Modern frameworks such as FEniCS UFL support automated adjoint derivation, code generation, and checkpointing for scalability and efficiency (Funke et al., 2013, Marin et al., 2018).
• All-at-once (full KKT) and penalty methods: Both the state and controls (and sometimes adjoints) are updated simultaneously, leading to large saddle-point systems. This is feasible in moderate-scale settings but problematic for very large problems (Leeuwen et al., 2015).
• Augmented and interior-point approaches: For problems with control or state box constraints, $\ell^1$-regularization, or variational inequalities, semismooth Newton or interior-point methods have been developed. These require nonsmooth analysis, active-set or log-barrier regularization, and specialized preconditioners for saddle-point systems (Porcelli et al., 2016, Pearson et al., 2018, Hartland et al., 2024).
• ADMM and splitting for nonsmooth or composite objectives: Operator splitting (ADMM) allows decoupling the smooth PDE-constrained block from nonsmooth regularization (such as TV, box, or sparsity terms), solving each subproblem with the most effective tool (e.g., PINN for PDE, soft-thresholding for $\ell^1$), and offering mesh-free, easily extensible algorithms (Song et al., 2023).
• Spectral and high-order discretizations: Matrix-free spectral element or pseudospectral methods efficiently handle high-order and unsteady PDEs at scale, integrated with automatic transpose operators for adjoint application and checkpointing (Marin et al., 2018, Aduamoah et al., 2020).

3. Learning, Operator Surrogates, and Modern Extensions

Recent advances incorporate machine learning, neural operators, and distributional perspectives:

• Physics-informed operator learning: DeepONet, Fourier Neural Operator (FNO), and related architectures enable training surrogate models that learn the mapping from control/design space to PDE solutions—or even directly to optimal controls or observables. Crucial for PDE-constrained optimization, recent results show that Fréchet-derivative approximation is necessary for accurate optimization, motivating derivative-informed losses during training (Wang et al., 2021, Yao et al., 16 Dec 2025, Cheng et al., 16 Jun 2025).
• Derivative-invariant architectures: Algorithms such as Virtual-Fourier neural operators and DIFNOs enable simultaneous approximation of solution operators and their sensitivities, enhancing stability and efficiency of surrogate-driven optimization (Cheng et al., 16 Jun 2025, Yao et al., 16 Dec 2025).
• Generative neural parameterization: Moving beyond single minimizers, generative reparameterization via latent-variable neural networks allows learning a map from random noise to an ensemble of near-optimal controls—enabling efficient exploration of multimodal or distributional solution spaces in settings with multiple (nearly degenerate) optima (Joglekar, 2024).
• Bi-level PINN architectures and hypergradient computation: Two-level approaches decouple the PDE solve (via PINN inner optimization) from control optimization, rigorously differentiating through the implicit solver using hypergradients. Broyden's method efficiently approximates the required Jacobian-inverse-vector products for large-scale problems (Hao et al., 2022).
• Learning-accelerated surrogates: Dual-network surrogate frameworks pair a learned dynamic predictor (e.g., time-discrete neural operator) with a proxy optimizer for the control, enabling near real-time solutions in control and design settings, with robust primal–dual training ensuring feasibility and constraint satisfaction (Guven et al., 29 Sep 2025).

4. Special Topics: Nonsmooth Constraints, Mesh Optimization, Stochasticity, and Quantum Methods

PDE-constrained optimization encompasses several specialized and emerging subfields:

• Sparse, nonsmooth, and control-constrained problems: Incorporation of $\ell^1$ penalties, total variation, and box constraints necessitates semismooth Newton methods, interior-point frameworks, or ADMM-PINN hybrid schemes. Robust active-set or saddle-point preconditioners are critical for achieving mesh- and parameter-independent performance in large-scale discretizations (Porcelli et al., 2016, Pearson et al., 2018, Song et al., 2023).
• High-order mesh adaptivity: Mesh optimization frameworks couple geometric metrics (e.g., Target-Matrix Optimization Paradigm) with PDE-constrained error measures, using adjoint sensitivity analysis to compute exact gradients with respect to mesh coordinates, and applying convolutional (Helmholtz) regularization for stability in high-order, unstructured, multi-element meshes (Kolev et al., 2 Jul 2025).
• Stochastic PDE-constrained optimization: Parameter uncertainty or random field inversion is approached by treating the parameter law as an optimization variable in $\mathcal{P}(\Theta)$ (space of probability measures), seeking to “push forward” the parameter law to match the observed data law. Wasserstein and KL-based gradient flows in parameter space, discretized via particle or JKO schemes, extend PDE-constrained optimization to distributional settings (Li et al., 2023).
• Quantum algorithms: Block-encoding and quantum Hamiltonian descent-based solvers have been developed to achieve exponential or polynomial quantum speedup in high-dimensional or high-resolution PDE-constrained optimization problems, by composing quantum PDE solvers (e.g., LCHS) and block-encoded objective oracles, and avoiding classical readout overheads (Sato et al., 18 Nov 2025).

5. Applications and Validation

PDE-constrained optimization is applied across a wide range of scientific and engineering problems:

• Optimal control and design: Navier–Stokes optimal control, optimal turbine placement in hydrodynamics, tsunami boundary data assimilation, and meshing for elasticity or Poisson equations (Funke et al., 2013, Kolev et al., 2 Jul 2025).
• Inverse modeling in medical imaging: Image registration, tumor growth model/data assimilation, and cardiovascular data assimilation leverage PDE-constrained regularization, Gauss–Newton–Krylov solvers, and high-performance spectral or adaptive FE discretizations (Mang et al., 2018).
• Physical-layer deep learning: Neural closure models in turbulence, segmentation models in bioimaging with PDE-priors, and parameterized surrogate operators for scientific ML all rely on embedded PDE constraints for inductive bias, regularization, and physical interpretability (Sirignano et al., 2021, Poudel et al., 1 Feb 2026).

6. Algorithmic Scalability, Performance, and Limitations

The scaling and performance characteristics of PDE-constrained optimization frameworks are determined by discrete adjoint evaluation (forward/adjoint cost ratio), the efficacy of checkpointing, preconditioning in saddle-point systems, and (for learning-based approaches) the accuracy of surrogate derivatives:

Approach	Scaling Behavior	Typical Bottleneck	Performance Example
Adjoint-gradient (FEniCS, PETSc/TAO)	Forward:Adjoint cost 1–2x	Nonlinear PDE solves; memory (adjoint tape)	$10^{8}$–$10^{9}$ DOF on 10⁴+ cores (Marin et al., 2018, Funke et al., 2013)
Interior-point, semismooth Newton	Mesh- and parameter robust	KKT system size, preconditioning	10–20 Krylov iterations, up to $10^7$ DOF (Porcelli et al., 2016, Hartland et al., 2024)
Surrogate learning (DeepONet, FNO)	Once-trained: $\sim$0.1–1s per forward/gradient eval	Surrogate accuracy, memory for derivative	$10^4$–$10^5$ control parameters, $100$–$1000\times$ speedup (Wang et al., 2021, Yao et al., 16 Dec 2025)

Limitations across the field include the curse of dimensionality in parameter/uncertainty quantification, sensitivity to training distribution for learned surrogates, and the challenge of enforcing strict feasibility or robust performance in black-box learning architectures (Funke et al., 2013, Yao et al., 16 Dec 2025, Wang et al., 2021, Li et al., 2023).

7. Perspectives and Future Directions

Research continues to develop scalable and robust PDE-constrained optimization engines for new regimes:

• Higher-fidelity learning: Derivative-informed neural operators with reduced-basis or mixed-resolution approaches significantly reduce sample and memory complexity for high-dimensional inference (Yao et al., 16 Dec 2025).
• Distributional and multi-modal inverse design: Generative neural network reparameterizations enable simultaneous exploration of multiple local minima, furthering the goals of distributional optimization in complex physical systems (Joglekar, 2024).
• Mesh-free, physics-informed methodologies: PINN and bi-level architectures allow enforced satisfaction of PDEs without explicit meshing, central for scientific ML and data-driven discovery (Hao et al., 2022, Song et al., 2023).
• Stochastic, uncertain, or robust formulations: Push-forward and optimal transport gradient flows extend to cases with stochastic parameters or partially observed data laws (Li et al., 2023).
• Quantum and exascale implementations: Leveraging quantum block-encodings and Hamiltonian descent, or distributed AMG and high-performance FE libraries, permits real-time, large-scale optimization for industrial and scientific challenges (Sato et al., 18 Nov 2025, Hartland et al., 2024).

PDE-constrained optimization continues to serve as a unifying language for design, control, inference, and scientific discovery, integrating advances from numerical PDEs, optimization theory, machine learning, scientific computation, and quantum information science.