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Optimal control of PDEs using physics-informed neural networks

Saviz Mowlavi, Saleh Nabi

arXiv:2111.09880v4math.OCphysics.comp-ph

TL;DR

The paper asks whether PINNs can solve PDE-constrained optimal-control problems while satisfying known governing equations and minimizing a cost objective. It proposes training and loss-weight guidelines, then compares PINN solutions with adjoint-based optimization across four PDE families. Both approaches produced comparable objective costs in high-fidelity evaluation, while the paper identifies implementation flexibility for PINNs alongside greater tuning demands and computational caveats.

  • Problem

    PDE-constrained optimal control requires finding cost-minimizing controls for fully known PDEs, but continuous control fields make direct search impractical and composite PINN losses can trade off PDE satisfaction against objective reduction.

  • Method

    The paper trains PINNs using forward-problem-informed architecture and parameter choices, tunes the critical cost weight with a two-step line search, and benchmarks against adjoint-based optimization.

  • Results

    Across Laplace, Burgers, Kuramoto-Sivashinsky, and Navier-Stokes examples, PINN and DAL solutions yielded comparable objective costs when evaluated with a separate high-fidelity numerical solver.

  • Takeaways & Limitations

    PINNs can be similarly effective as DAL while offering easier implementation and flexibility across governing equations, boundary conditions, geometries, and cost objectives.

  • Takeaways & Limitations

    PINN parameter tuning is more involved, while the DAL timing comparison is limited because calculations used a single CPU core.

Abstract

from arXiv · show

Physics-informed neural networks (PINNs) have recently become a popular method for solving forward and inverse problems governed by partial differential equations (PDEs). By incorporating the residual of the PDE into the loss function of a neural network-based surrogate model for the unknown state, PINNs can seamlessly blend measurement data with physical constraints. Here, we extend this framework to PDE-constrained optimal control problems, for which the governing PDE is fully known and the goal is to find a control variable that minimizes a desired cost objective. We provide a set of guidelines for obtaining a good optimal control solution; first by selecting an appropriate PINN architecture and training parameters based on a forward problem, second by choosing the best value for a critical scalar weight in the loss function using a simple but effective two-step line search strategy. We then validate the performance of the PINN framework by comparing it to adjoint-based nonlinear optimal control, which performs gradient descent on the discretized control variable while satisfying the discretized PDE. This comparison is carried out on several distributed control examples based on the Laplace, Burgers, Kuramoto-Sivashinsky, and Navier-Stokes equations. Finally, we discuss the advantages and caveats of using the PINN and adjoint-based approaches for solving optimal control problems constrained by nonlinear PDEs.

1. Introduction

The paper extends physics-informed neural networks to PDE-constrained optimal control, where known governing equations constrain the search for cost-minimizing controls. It proposes guidelines for training and evaluating PINN solutions and compares them with adjoint-based optimization across several PDE examples.

  • Motivation: PDE-constrained optimal control seeks control variables that minimize application-specific costs while satisfying fully known governing PDEs.The paper situates these problems in applications including fluid mechanics, heat transfer, electromagnetism, topology optimization, and mesh refinement.
  • Motivation: Adjoint-based optimization efficiently computes cost gradients for space- or time-dependent controls using only two PDE simulations.The introduction identifies this efficiency as a major reason adjoint-based methods are commonly used.
  • Contributions: The proposed PINN guidelines first select architecture and training parameters from a forward problem governed by the same PDE.This establishes a forward-solving baseline before training the optimal-control model.
  • Contributions: A two-step line search selects the scalar cost weight in the PINN loss by evaluating the resulting control with a separate forward PINN computation.The weight is critical because the solution must satisfy the PDE constraint while minimizing the cost objective.
  • Contributions: The framework is applied to Laplace, Burgers, Kuramoto-Sivashinsky, and Navier-Stokes optimal-control problems and compared with adjoint-based results.The paper presents this comparison as part of its evaluation of PINN optimal-control solutions.

2. Methodology

The methodology formulates PDE-constrained control as minimizing a cost functional subject to governing, boundary, and initial equations, then solves it with PINNs or adjoint-based optimization. PINNs jointly approximate state and control using a composite loss, while a scalar cost weight and training guidelines address the balance between objective minimization and PDE satisfaction.

  • 2.1. Optimal control problem statement: PDE-constrained optimal control seeks a control c that minimizes J(u,c) while the state satisfies the governing PDE, boundary conditions, and initial condition.The control may represent volume, boundary, or initial actuation, and the discretized state and control belong to Euclidean spaces.
  • 2.1. Optimal control problem statement: Continuous control fields make brute-force enumeration of candidate controls impractical because the search space is infinite-dimensional.The paper therefore introduces PINN-based optimization and uses adjoint-based optimization to provide benchmark solutions.
  • 2.2. Physics-informed neural networks for optimal control: A PINN represents the state u(x,t) with a fully connected neural network whose inputs are space and time and whose output approximates the state.The network uses trainable layer weights, biases, and nonlinear activation functions; automatic differentiation supplies derivatives for PDE-loss evaluation.
  • 2.2. Physics-informed neural networks for optimal control: The optimal-control PINN jointly trains state and control networks by minimizing an augmented loss containing PDE, boundary, initial-condition, and cost terms.The parameters of both networks are updated concurrently with gradient-based optimization until the trained networks approximately solve the constrained problem.
  • 2.2. Physics-informed neural networks for optimal control: The methodology uses Glorot initialization, tanh activations, Adam optimization, and normalized space-time inputs for the neural networks.These settings are stated as the choices used in the reported results.
  • 2.3. Adjoint-based optimal control: Adjoint-based optimization constructs a Lagrangian with an adjoint field, enforces the governing equation through stationarity, and obtains the total cost gradient with respect to control.At an optimal solution, the derivative of the cost with respect to the control is zero.
  • 2.4. Guidelines for training and evaluating the PINN optimal solution: The composite PINN loss can favor the cost objective over PDE satisfaction, or vice versa, because the PDE loss should vanish while the cost loss may remain finite.The paper addresses this challenge with guidelines that select architecture and training parameters from a forward problem, then search for an appropriate scalar cost weight.

3. Results

The study applies PINN and adjoint-based approaches to optimal control problems governed by four PDEs and compares the resulting controls using high-fidelity forward simulations. The comparison covers Laplace, Burgers, Kuramoto-Sivashinsky, and Navier-Stokes systems.

  • 3. Results: The experiments apply both PINN and adjoint-based approaches to Laplace, Burgers, Kuramoto-Sivashinsky, and Navier-Stokes optimal-control problems.For each PDE, the PINN architecture is first validated on a forward problem against a reference solution before solving the control problem.
  • 3. Results: The resulting controls are compared by evaluating their cost objectives with high-fidelity forward simulations using finite-element or spectral numerical methods.The high-fidelity solver computes the state for each fixed PINN or adjoint-based control to obtain an accurate estimate of J.

3.1. Laplace equation

The Laplace study first validates PINN settings on a forward problem, then applies the same framework to convex boundary optimal control and compares it with DAL.

  • Forward problem: The forward Laplace problem uses a four-hidden-layer network with 50 neurons per layer, residual and boundary training points, minibatches, and scheduled learning-rate decay.Training uses 10,000 residual points, 160 boundary points, an initial learning rate of 10^-3, and 6000 epochs.
  • Forward problem: Decreasing loss components track decreasing relative L2 test error, while the trained PINN closely matches the analytical solution and validates the selected architecture and training points.The test error is evaluated on an independent 100×100 Cartesian grid.
  • Optimal control problem: The optimal-control problem seeks a top-wall potential f*(x) that produces the desired flux qd(x) under Laplace dynamics with periodic lateral boundaries.The problem is convex because the Laplace equation is linear and the objective is quadratic.
  • Optimal control problem: PINNs solve the control problem with a second network for f(x), while the state network retains the forward architecture and both networks train simultaneously.The control network has three hidden layers of 30 neurons each.
  • Optimal control problem: The two-step line search selects wJ = 100 because its resulting forward solution produces the lowest cost objective J among the tested values.The selected PINN control is subsequently compared with the DAL solution.
  • Optimal control problem: Excellent agreement among the PINN, DAL, and analytical top-wall potentials indicates that both numerical frameworks found the global solution for this convex problem.The comparison uses the optimal potentials obtained from the two frameworks and the analytical counterpart.

3.2. Burgers equation

The Burgers experiments first validate PINN forward-solution accuracy against an analytical solution, then apply PINNs to a nonlinear optimal-control problem and compare them with DAL. The selected PINN control produces a final state nearly identical to the target, although multiple local minima and a smoothness trade-off remain.

  • Forward problem: The forward Burgers problem uses periodic boundaries, viscosity ν = 0.01, domain length L = 4, and horizon T = 5, with the analytical solution as the initial state.The PINN is evaluated against the analytical solution.
  • Forward problem: A four-hidden-layer PINN trained on residual, boundary, and initial points achieves excellent agreement with the analytical Burgers solution and decreasing relative L2 error.The agreement is shown through final-time snapshots, contour plots, and local absolute error.
  • Optimal control problem: The optimal-control problem seeks an initial condition whose evolved state matches the analytical final state, but Burgers nonlinearity permits multiple local minima.The analytical initial condition is the global-minimum reference, while other controls may produce similar final states.
  • Optimal control problem: The PINN jointly trains a state network and an initial-condition network, while DAL iteratively solves forward and adjoint Burgers equations and updates the control by gradient descent.The two approaches use different optimization formulations for the same control objective.
  • Optimal control problem: The two-step line search selects wJ = 1 because its control yields the lowest separately estimated cost objective among the tested weights.The selected solution is then compared with the DAL solution.
  • Optimal control problem: The PINN and DAL controls produce final states almost identical to the analytical target, with J = 2.27 · 10^-7 for PINN and J = 7.12 · 10^-8 for DAL.DAL attains the slightly lower cost, while both methods identify initial conditions substantially different from the analytical initial state.

3.3. Kuromoto-Sivashinsky equation

The KS example tests PINN optimal control in a chaotic regime, using a forward solve to validate the network before comparing PINN and DAL control solutions. A two-step search selects the loss weight, and both methods drive the state near zero with nearly identical costs.

  • Problem: At L = 50, the unforced KS system is chaotic, so the control force is designed to drive its state toward the unstable zero fixed point.The zero state is stable for L < 2π but unstable for L > 2π; larger domains can become chaotic.
  • Forward problem: The forward PINN uses a 5-layer, 50-neuron architecture with 80000 residual points, 82 boundary points, and 41 initial points.The trained solution shows excellent agreement with a spectral reference at T = 10, validating these choices.
  • Cost design: Regularizing the control norm prevents the optimizer from choosing an unbounded force concentrated immediately after the initial time.The quadratic cost penalizes both the state and control-force norms.
  • Optimal control method: The optimal-control PINN represents both the state and distributed force with neural networks trained simultaneously using the validated architecture and training points.The cost integral is evaluated by Monte Carlo integration using the same residual-point minibatches.
  • Weight selection: The two-step line search selects wJ = 10^-3 because its separately evaluated forward solution yields the lowest cost objective.Step 1 examines loss components across weights; Step 2 evaluates the resulting fixed-control solutions.
  • Results: J = 20.58 and J = 20.64 for PINN and DAL controls, respectively, while both spectral forward solutions drive the KS state toward near-zero values.The two optimal forcing fields look very similar in the reported comparisons.

3.4. Navier-Stokes equations

The Navier–Stokes example first validates a PINN against OpenFOAM, then optimizes the inlet profile to recover a near-parabolic outlet. PINN and DAL produce different inlet profiles but comparable outlet costs.

  • Problem setup: The steady 2D incompressible Navier–Stokes problem uses Re = 100 with prescribed inlet, blowing, suction, outflow, and no-slip wall conditions.The setup includes uniform velocity 0.3 at the blowing and suction boundaries and a reference pressure of zero.
  • Forward problem: The forward PINN uses one 5-layer, 50-neuron network for velocity and pressure and is compared with a finite-volume OpenFOAM solution.The outlet velocity profiles show very good agreement between the trained PINN and OpenFOAM solutions.
  • Boundary conditions: Pressure boundary conditions are included in the PINN loss because they affect the resulting velocity profile and are needed for comparison with OpenFOAM.The authors note that pressure conditions are often treated as unnecessary in PINN formulations when unknown.
  • Optimal control problem: The control problem optimizes the inlet profile so the outlet profile approaches the target parabolic profile observed in the forward problem.The parabolic inlet produced a skewed outlet profile, motivating inlet-profile control.
  • Optimal control method: The PINN adds a 3-layer, 30-neuron control network and trains it jointly with the flow network across 12 values of wJ from 10^-3 to 10^5.The control network is trained with the same flow training points and a midpoint-rule cost evaluation.
  • Results: J = 0.00278 and J = 0.00265 for PINN and DAL inlet profiles, respectively, and both produce near-parabolic outlet profiles.The PINN inlet is smoother and yields an outlet profile with a more parabolic shape, while sharper DAL features do not produce a noticeably lower cost.

4. Discussion

Across four PDE-constrained control examples, PINNs and DAL produced comparable high-fidelity costs, while nonlinear cases revealed trade-offs in smoothness and computational setup. PINNs were easier to implement, but computational comparisons remain non-rigorous and parameter tuning is more involved for PINNs.

  • Both frameworks returned comparable objective costs when evaluated with a separate high-fidelity numerical solver, although PINNs required an extra evaluation step.
  • The Laplace problem had a global analytical optimum recovered by both methods, while nonlinear examples exposed differences in the resulting control distributions.
  • DAL produced lower-cost but less-smooth controls for Burgers and Navier–Stokes, whereas PINNs produced smoother controls.
  • PINNs offer easier implementation and flexibility across governing equations, boundary conditions, geometries, and cost objectives than the manually derived and implemented DAL workflow.
  • Computational efficiency cannot be compared rigorously because PINN and DAL costs depend on different hardware, software, optimization, discretization, and architecture choices.
  • PINNs may become relatively faster for more complex problems, but DAL timings used a single CPU core, PINNs require more parameter tuning, and alternative adjoint algorithms may alter the comparison.

5. Conclusions

The paper proposes PINN guidelines for PDE-constrained optimal control and compares them with DAL across four equations of increasing complexity. Both approaches achieved comparable high-fidelity costs, while PINNs offered flexibility and easier implementation but DAL remained advantageous in computational cost in some settings.

  • The study proposes a methodology and guidelines for solving PDE-constrained optimal control problems with PINNs.
  • PINNs and DAL were compared across Laplace, Burgers, Kuramoto-Sivashinsky, and Navier–Stokes examples with increasing complexity.
  • Both techniques found controls with comparable objective costs after evaluation in a high-fidelity numerical solver.
  • PINNs provide flexibility and ease of implementation, whereas DAL can offer lower computational cost in some settings.

Appendix A. Adjoint Burgers equation

The appendix derives the adjoint Burgers equation and its control gradient from the general adjoint methodology. The forward field is coupled to an adjoint field with periodic boundaries and a specified terminal condition.

  • The adjoint Burgers equation and cost gradient with respect to the initial-condition control are derived from the general adjoint methodology.
  • The adjoint field λ(x, t) accompanies the forward Burgers field u(x, t) generated by the controlled initial condition u0(x).
  • The adjoint problem uses periodic boundary conditions and terminal condition λ(x, T) = −u + ud.

Appendix B. Adjoint Kuramoto-Sivashinsky equation

The appendix derives the adjoint Kuramoto–Sivashinsky equation and the gradient with respect to the forcing control. The forward and adjoint fields are linked through periodic boundaries and a zero terminal condition.

  • The adjoint Kuramoto–Sivashinsky equation and cost gradient with respect to the forcing are derived using the general adjoint methodology.
  • The adjoint field λ(x, t) accompanies the forward KS field u(x, t) generated by the forcing control f(x, t).
  • The adjoint KS problem uses periodic boundary conditions and terminal condition λ(x, T) = 0.

Appendix C. Adjoint Navier-Stokes equations

The appendix derives the adjoint formulation for the Navier–Stokes control problem by varying an augmented Lagrangian and enforcing stationarity. It obtains adjoint equations, boundary conditions, and the control gradient.

  • Lagrangian formulation: The Navier–Stokes control problem is expressed through an augmented objective functional, or Lagrangian, using Einstein notation for the velocity field.The velocity and adjoint fields are represented componentwise in the appendix.
  • Variational derivation: Setting the Lagrangian variation to zero enables derivation of the Euler–Lagrange equations using vector calculus and integration by parts.The derivation separates volumetric and surface-integral contributions.
  • Adjoint boundary conditions: The resulting surface terms determine boundary conditions for the adjoint velocity and pressure fields on the specified boundary segments.The appendix states conditions including λ1 = λ2 = 0 and (n · ∇)Π = 0 on Γi ∪ Γb ∪ Γs ∪ Γw.
  • Control gradient: The appendix concludes with the total gradient of the cost objective with respect to the control, termed the design equation.The stated gradient expression is evaluated using values at the inlet.
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