Projected Adjoint-Based Methods

• Projected adjoint-based methods are advanced techniques that use adjoint equations, modal projections, and operator inference to compute gradients for PDE-constrained optimization.
• They integrate methods like Proper Orthogonal Decomposition and the Dynamical Arnoldi Method for reduced-order modeling, offering scalability and robustness against noisy and sparse data.
• The approach enforces state constraints by projecting gradients onto the constraint manifold via dual adjoint solves, ensuring accurate optimization with temporal regularization.

Projected adjoint-based methods are a class of techniques in applied mathematics and computational science that leverage adjoint equations, modal projections, and operator-inference frameworks to efficiently compute gradients or enforce constraints, particularly in the context of PDE-constrained optimization and reduced-order modeling. These methods are designed to address both the computational challenges of high-dimensional systems and the robustness issues arising from noisy or sparse data, as well as general state constraints. Key approaches integrate continuous-time adjoint analysis, Galerkin-type modal reductions, and manifold projections within optimization loops.

1. Fundamental Principles and Methodologies

Projected adjoint-based methods exploit the adjoint-state formalism to facilitate gradient computation and constraint enforcement in infinite- and finite-dimensional settings. Key methodological features include:

• Adjoint-State Equations: The adjoint variable $\lambda(t)$ or $q^*$ arises from imposing stationarity of the Lagrangian with respect to primal variables. For a dynamical system $\dot{u}=F(u,s)$ or a PDE $B(q) = 0$, linearization yields a dual evolution equation (backward in time) for adjoint variables, typically of the form:

$\partial_t q^* = -A^T q^* - g,\quad q^*(T) = 0$

where $A$ represents the Jacobian of the primal operator (Reiss et al., 2018).

• Projection onto Modal Subspaces: Proper Orthogonal Decomposition (POD) or Krylov/Arnoldi-based modal decompositions (such as via the Dynamical Arnoldi Method, DAM) are employed to construct low-rank bases $V_m$ approximating the dominant modes of the linearized operator. Primal and adjoint equations are projected into these subspaces, yielding reduced-order systems for computational efficiency (Liu et al., 12 Jan 2026, Reiss et al., 2018).
• Trajectory-Based Loss and Gradient: In reduced-order modeling, the objective function is defined as an $L_2$-in-time misfit between projected model states and measured data,

$$J(\theta) = \frac{1}{2} \int_0^T \|q(t;\theta) - q_{\rm true}(t)\|_2^2\,dt$$

where $q(t;\theta)$ solves the reduced-order ODE, and $q_{\rm true}(t)$ is the projection of state trajectories (Liu et al., 12 Jan 2026).

• Enforcement of State Constraints: For PDE optimization with general state constraints $G(y)=0$, a principled projection of the unconstrained adjoint-based gradient onto the tangent space of the constraint manifold is computed using a second adjoint equation with data-dependent forcing. The projected gradient is formulated as

$$P_u(\nabla_u J) = \nabla_u J - \mathcal{K}_u^*(\mathcal{K}_u \mathcal{K}_u^*)^{-1} \mathcal{K}_u(\nabla_u J)$$

where $\mathcal{K}_u$ encodes the constraint-Jacobian composition (Matharu et al., 2023).

2. Modal Projections and Dynamical Arnoldi Method

Model reduction in projected adjoint-based approaches is anchored in constructing low-dimensional invariant subspaces:

• Arnoldi and DAM: Traditional Arnoldi factorization generates a basis $V_m$ for the Krylov subspace of the primal operator. The Dynamical Arnoldi Method (DAM) generalizes this by allowing flexible calculation plans, including nonstandard vector choices and field modifications, to target non-symmetric or coupled systems inaccessible to classic Arnoldi. The spatial operator $A$ is approximated as $A \approx V_m H_m V_m^T$ with $H_m$ upper Hessenberg (Reiss et al., 2018).
• Projected Adjoint Equation: The adjoint variable is expanded $q^* \approx V_m p$, and the adjoint ODE is projected,

$\dot{p} = -H_m^T p - V_m^T g$

enabling efficient backward integration and gradient assembly entirely within the reduced subspace (Reiss et al., 2018).

These projections ensure that gradient and constraint information are efficiently transferred between the full and reduced-order systems while maintaining numerical tractability for high-dimensional problems.

3. Projected Adjoint Optimization in Reduced-Order Modeling

In operator inference and nonlinear ROM training, projected adjoint-based methods present an alternative to classical regression on finite-difference data:

• Continuous-Time Operator Inference: The reduced ODE $\dot{q}(t) = f(q(t); \theta)$ is matched to projected measurement data via trajectory loss, specifically avoiding noisy numerical differentiation.
• Adjoint-Based Gradient Computation: The gradient of $J(\theta)$ is obtained by solving the forward reduced system for $q(t)$, the backward adjoint problem for $\lambda(t)$,

$$\dot{\lambda}(t) = -[\partial_q f(q(t); \theta)]^T \lambda(t) - [q(t) - q_{\rm true}(t)], \quad \lambda(T) = 0$$

and then assembling

$$\frac{\partial J}{\partial \theta} = \int_0^T \lambda(t)^T \frac{\partial f(q(t); \theta)}{\partial \theta}dt$$

(Liu et al., 12 Jan 2026).

• One-Shot Algorithm: Each optimization iteration entails one forward integration, one backward adjoint solve, gradient assembly, and parameter update. This yields computational cost independent of parameter dimension $d$ and reduces sensitivity to temporal discretization and measurement noise.
• Temporal Regularization: By fitting entire trajectories, the method introduces intrinsic temporal smoothing, improving robustness under sparse or noisy sampling relative to traditional finite-difference approaches.

4. Projected Adjoint Enforcement of State Constraints

Projected adjoint methods address PDE-constrained optimization with state constraints $G(y)=0$ via a two-adjoint solve strategy:

• Unconstrained Gradient: The classical adjoint provides $\nabla_u J$.
• Constraint Projection: The constraint tangent space is characterized by the nullspace of the composed linear mapping $\mathcal{K}_u$. Projection is realized by solving for $\lambda$ in the small system $$(\mathcal{K}_u\mathcal{K}_u^*)\,\lambda = \mathcal{K}_u(\nabla_u J)$$, then solving a second adjoint PDE with right-hand-side $dG(y)^*\lambda$. The projected gradient is thus

$$d^{(n)} = g^{(n)} - E_u(y^{(n)}, u^{(n)})^* q^{(n)}$$

where $q^{(n)}$ solves the projection adjoint equation (Matharu et al., 2023).

• Algorithmic Structure: The iterative algorithm alternates between state PDE solves, primary and projection adjoint computations, projected gradient formation, and update steps. Retractions onto the manifold $M$ can be incorporated where feasible, but are not required for approximate constraint enforcement.

5. Computational and Empirical Insights

Projected adjoint-based methods offer salient computational properties:

• Complexity Analysis: For reduced-order modeling, each iteration costs $O(N_t r^3)$ for forward and backward solves, and $O(N_t r d)$ for gradient accumulation, per iteration. This is contrasted with $O(d^2 k)$ for classical OpInf regression, enabling scalability to large parameter spaces (Liu et al., 12 Jan 2026).
• Empirical Validation:
  • In 1D and 2D PDE benchmarks (e.g., viscous Burgers', Fisher–KPP, advection–diffusion), adjoint-based ROM training consistently outperforms finite-difference-based methods under sparse samplings (as few as 20 snapshots) and high noise (up to 200% of signal standard deviation), achieving stable roll-out and lower relative state errors (Liu et al., 12 Jan 2026).
  • In state-constrained PDE optimization (1D heat, 2D Navier–Stokes closure), the projected adjoint enforces constraints within $O(\tau^2)$ drift and preserves cost reduction, matching analytic gradients with second-order accuracy under discretization (Matharu et al., 2023).
  • DAM-based modal adjoint projection achieves machine-precision error in simple cases and ~1% relative error in more complex settings, with proper calculation plan tuning required for coupled or non-symmetric systems (Reiss et al., 2018).

6. Applications and Generalizations

Projected adjoint-based methods find applications in:

• Data-driven reduced-order modeling: Construction and robust training of ROMs from high-dimensional dynamical systems, including systems with sparse or noisy data.
• PDE-constrained optimization: Efficient and regular optimization for control, inverse design, and closure modeling, especially under state or energy constraints.
• Fluid dynamics and control: Efficient adjoint solutions for noise cancellation, closure calibration, and control synthesis without hand-coding discrete adjoints.
• High-codimension and nonlinear constraint management: By stacking multiple adjoint solves, the approach generalizes to constraints of arbitrary codimension.

Potential extensions include handling inequality constraints, trust-region adaptations to mitigate off-manifold drift, and improved global convergence strategies for high-dimensional or highly nonlinear constraints (Matharu et al., 2023).

7. Advantages, Limitations, and Outlook

Advantages of projected adjoint-based methods:

• Gradient and constraint projection without explicit Lagrange multipliers or saddle-point systems
• Computational efficiency: Modal projections, single forward–backward solves per iteration, independence from parameter dimension scaling, and no need for full Jacobian assembly.
• Robustness to data sparsity and noise: Intrinsic temporal smoothing and projection-based regularization outperform standard methods under adverse data regimes.
• Flexibility: DAM and general projection operators adapt to varied system structures and state constraints.

Limitations and open challenges:

• Only $O(\tau^2)$-accurate constraint enforcement without explicit retraction; exact invariance may require problem-specific mappings.
• Increased per-iteration cost under multiple or nonlinear constraints due to solving additional adjoint systems.
• **Potential ill-conditioning when constraint Jacobians are highly nonlinear, requiring stabilization in $(\mathcal{K}_u\mathcal{K}_u^*)^{-1}$ solves.
• Lack of built-in handling for inequality constraints, necessitating the development of smoothing, regularization, or active-set strategies for broader applicability.

Projected adjoint-based methods thus represent a rigorously grounded, scalable, and practical toolkit for data-driven scientific computing and complex PDE-constrained optimization, particularly well-suited to contemporary high-dimensional, data-limited, or constraint-rich scenarios (Liu et al., 12 Jan 2026, Reiss et al., 2018, Matharu et al., 2023).