Hangrui Yue

Ming-Chih Lai, Yongcun Song, Xiaoming Yuan et al.

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 or with a high degree of accuracy, and they are decoupled from the learning of the PDEs. Its efficiency is promisingly validated by some elliptic and parabolic interface optimal control problems.

Yongcun Song, Xiaoming Yuan, Hangrui Yue

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.

Yongcun Song, Xiaoming Yuan, Hangrui Yue et al.

We propose an optimization-informed deep neural network approach, named iUzawa-Net, aiming for the first solver that enables real-time solutions for a class of nonsmooth optimal control problems of linear partial differential equations (PDEs). The iUzawa-Net unrolls an inexact Uzawa method for saddle point problems, replacing classical preconditioners and PDE solvers with specifically designed learnable neural networks. We prove universal approximation properties and establish the asymptotic $\varepsilon$-optimality for the iUzawa-Net, and validate its promising numerical efficiency through nonsmooth elliptic and parabolic optimal control problems. Our techniques offer a versatile framework for designing and analyzing various optimization-informed deep learning approaches to optimal control and other PDE-constrained optimization problems. The proposed learning-to-control approach synergizes model-based optimization algorithms and data-driven deep learning techniques, inheriting the merits of both methodologies.

Yongcun Song, Xiaoming Yuan, Hangrui Yue

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.

Ziqian Li, Kang Liu, Yongcun Song et al.

Operator learning has emerged as a promising paradigm for developing efficient surrogate models to solve partial differential equations (PDEs). However, existing approaches often overlook the domain knowledge inherent in the underlying PDEs and hence suffer from challenges in capturing temporal dynamics and generalization issues beyond training time frames. This paper introduces a deep neural ordinary differential equation (ODE) operator network framework, termed NODE-ONet, to alleviate these limitations. The framework adopts an encoder-decoder architecture comprising three core components: an encoder that spatially discretizes input functions, a neural ODE capturing latent temporal dynamics, and a decoder reconstructing solutions in physical spaces. Theoretically, error analysis for the encoder-decoder architecture is investigated. Computationally, we propose novel physics-encoded neural ODEs to incorporate PDE-specific physical properties. Such well-designed neural ODEs significantly reduce the framework's complexity while enhancing numerical efficiency, robustness, applicability, and generalization capacity. Numerical experiments on nonlinear diffusion-reaction and Navier-Stokes equations demonstrate high accuracy, computational efficiency, and prediction capabilities beyond training time frames. Additionally, the framework's flexibility to accommodate diverse encoders/decoders and its ability to generalize across related PDE families further underscore its potential as a scalable, physics-encoded tool for scientific machine learning.