7 research outputs found
Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problems
We consider a general class of nonsmooth optimal control problems with
partial differential equation (PDE) constraints, which are very challenging due
to its nonsmooth objective functionals and the resulting high-dimensional and
ill-conditioned systems after discretization. We focus on the application of a
primal-dual method, with which different types of variables can be treated
individually and thus its main computation at each iteration only requires
solving two PDEs. Our target is to accelerate the primal-dual method with
either larger step sizes or operator learning techniques. For the accelerated
primal-dual method with larger step sizes, its convergence can be still proved
rigorously while it numerically accelerates the original primal-dual method in
a simple and universal way. For the operator learning acceleration, we
construct deep neural network surrogate models for the involved PDEs. Once a
neural operator is learned, solving a PDE requires only a forward pass of the
neural network, and the computational cost is thus substantially reduced. The
accelerated primal-dual method with operator learning is mesh-free, numerically
efficient, and scalable to different types of PDEs. The acceleration
effectiveness of these two techniques is promisingly validated by some
preliminary numerical results
A numerical approach to the optimal control of thermally convective flows
The optimal control of thermally convective flows is usually modeled by an
optimization problem with constraints of Boussinesq equations that consist of
the Navier-Stokes equation and an advection-diffusion equation. This optimal
control problem is challenging from both theoretical analysis and algorithmic
design perspectives. For example, the nonlinearity and coupling of fluid flows
and energy transports prevent direct applications of gradient type algorithms
in practice. In this paper, we propose an efficient numerical method to solve
this problem based on the operator splitting and optimization techniques. In
particular, we employ the Marchuk-Yanenko method leveraged by the
projection for the time discretization of the Boussinesq equations so
that the Boussinesq equations are decomposed into some easier linear equations
without any difficulty in deriving the corresponding adjoint system.
Consequently, at each iteration, four easy linear advection-diffusion equations
and two degenerated Stokes equations at each time step are needed to be solved
for computing a gradient. Then, we apply the Bercovier-Pironneau finite element
method for space discretization, and design a BFGS type algorithm for solving
the fully discretized optimal control problem. We look into the structure of
the problem, and design a meticulous strategy to seek step sizes for the BFGS
efficiently. Efficiency of the numerical approach is promisingly validated by
the results of some preliminary numerical experiments
The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approach
We study the combination of the alternating direction method of multipliers
(ADMM) with physics-informed neural networks (PINNs) for a general class of
nonsmooth partial differential equation (PDE)-constrained optimization
problems, where additional regularization can be employed for constraints on
the control or design variables. The resulting ADMM-PINNs algorithmic framework
substantially enlarges the applicable range of PINNs to nonsmooth cases of
PDE-constrained optimization problems. The application of the ADMM makes it
possible to untie the PDE constraints and the nonsmooth regularization terms
for iterations. Accordingly, at each iteration, one of the resulting
subproblems is a smooth PDE-constrained optimization which can be efficiently
solved by PINNs, and the other is a simple nonsmooth optimization problem which
usually has a closed-form solution or can be efficiently solved by various
standard optimization algorithms or pre-trained neural networks. The ADMM-PINNs
algorithmic framework does not require to solve PDEs repeatedly, and it is
mesh-free, easy to implement, and scalable to different PDE settings. We
validate the efficiency of the ADMM-PINNs algorithmic framework by different
prototype applications, including inverse potential problems, source
identification in elliptic equations, control constrained optimal control of
the Burgers equation, and sparse optimal control of parabolic equations
The Hard-Constraint PINNs for Interface Optimal Control Problems
We show that the physics-informed neural networks (PINNs), in combination
with some recently developed discontinuity capturing neural networks, can be
applied to solve optimal control problems subject to partial differential
equations (PDEs) with interfaces and some control constraints. The resulting
algorithm is mesh-free and scalable to different PDEs, and it ensures the
control constraints rigorously. Since the boundary and interface conditions, as
well as the PDEs, are all treated as soft constraints by lumping them into a
weighted loss function, it is necessary to learn them simultaneously and there
is no guarantee that the boundary and interface conditions can be satisfied
exactly. This immediately causes difficulties in tuning the weights in the
corresponding loss function and training the neural networks. To tackle these
difficulties and guarantee the numerical accuracy, we propose to impose the
boundary and interface conditions as hard constraints in PINNs by developing a
novel neural network architecture. The resulting hard-constraint PINNs approach
guarantees that both the boundary and interface conditions can be satisfied
exactly and they are decoupled from the learning of the PDEs. Its efficiency is
promisingly validated by some elliptic and parabolic interface optimal control
problems