Organizations: Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, 637371, Singapore. · Department of Mathematics, The University of Hong Kong, Pok Fu Lam, Hong Kong SAR, China. · School of Mathematical Sciences, Nankai University, Tianjin 300071, China.
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 ε-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.
Figures & tables
Figure 2.1 : The neural network architecture of QSk .
Figure 2.2 : The neural network architecture of QAk .
Figure 2.3 : An overview of the iUzawa-Net ( 1.7 ).
m
# of active ui∗
∣A(ui∗)∣/∣Ω∣ for active ui∗
Training
Testing
Training Mean
Testing Mean
64
11450
1422
0.2116
0.2128
128
—
1419
—
0.2130
256
—
1433
—
0.2256
Table 5.1: Statistics of active optimal control and active sets in the training and testing datasets of different resolutions m . A dash (—) indicates that the corresponding data is not applicable.
Method
m
εrel(u^)
εabs(u^)
Mean
SD
Mean
SD
iUzawa-Net-F
64
2.00×10−3
0.97×10−3
4.05×10−3
1.85×10−3
128
2.29×10−3
0.99×10−3
4.60×10−3
1.76×10−3
256
2.38×10−3
0.85×10−3
4.99×10−3
2.13×10−3
iUzawa-Net-S
64
3.67×10−3
3.02×10−3
7.03×10−3
3.74×10−3
128
3.73×10−3
2.79×10−3
7.15×10−3
3.69×10−3
Table 5.2: Comparison of numerical results for solving ( 5.1 ).
Figure 5.1 : Computed optimal control for a single instance of ( 5.1 ) with m=64 .
m
SSN
Primal-Dual
Inexact Uzawa
POD
iUzawa-Net-F
iUzawa-Net-S
64
79.2
25.1
42.7
1.1
0.3
0.5
128
176.7
218.7
177.1
3.8
0.4
0.6
256
773.9
966.0
355.0
18.3
0.6
1.0
Table 5.3: Computation time (milliseconds) of the SSN method, the primal-dual method, the inexact Uzawa method, the POD method, the iUzawa-Net-F , and the iUzawa-Net-S for solving ( 5.1 ) under different resolutions m (averaged over instances in the testing sets).
m
SSN
Primal-Dual
Inexact Uzawa
POD
iUzawa-Net-F
iUzawa-Net-S
64
2.59
10.77
17.94
2.36
6
10
128
1.91
10.78
7.71
2.43
6
10
256
1.94
10.78
6.57
2.60
6
10
Table 5.4: Number of iterations or layers of the SSN method, the primal-dual method, the inexact Uzawa method, the POD method, the iUzawa-Net-F , and the iUzawa-Net-S for solving ( 5.1 ) under different resolutions m (averaged over instances in the testing sets).
iUzawa-Net-F
iUzawa-Net-S
L
Parameters
εrel(u^)
L
Parameters
εrel(u^)
Mean
SD
Mean
SD
2
90,698
6.95×10−3
4.53×10−3
2
45,349
1.21×10−2
1.12×10−2
3
136,047
4.14×10−3
2.38×10−3
3
45,349
6.15×10−3
2.95×10−3
6
272,094
2.00×10−3
0.97×10−3
6
45,349
5.88×10−3
3.21×10−3
10
453,490
1.63×10−3
0.77×10−3
10
45,349
3.67×10−3
3.02×10−3
Table 5.5: Relative errors εrel(u^) on the testing set with m=64 of the iUzawa-Net-F and the iUzawa-Net-S with different numbers of layers L for solving ( 5.1 ).
Figure 5.2 : Computed optimal control for a single instance of ( 5.3 ) with m=64 .
Method
m
εrel(u^)
εabs(u^)
Mean
SD
Mean
SD
iUzawa-Net-F
64
7.14×10−3
1.02×10−2
1.85×10−2
1.82×10−2
128
5.52×10−3
4.48×10−3
1.50×10−2
1.10×10−2
256
6.89×10−3
6.68×10−3
1.69×10−2
1.36×10−2
iUzawa-Net-S
64
4.71×10−3
5.40×10−3
1.12×10−2
8.81×10−3
128
4.12×10−3
2.96×10−3
1.02×10−2
7.72×10−3
Table 5.6: Numerical results for solving ( 5.3 ).
m
SSN
Primal-Dual
Inexact Uzawa
POD
iUzawa-Net-F
iUzawa-Net-S
64
4.29×103
1.14×103
171.1
0.8
0.3
0.5
128
2.14×104
8.71×103
1.17×103
2.2
0.4
0.6
256
6.07×104
2.30×104
4.94×103
9.2
0.6
1.0
Table 5.7: Computation time (milliseconds) of the SSN method, the primal-dual method, the inexact Uzawa method, the POD method, the iUzawa-Net-F , and the iUzawa-Net-S for solving ( 5.3 ) with different grid resolutions m (averaged over instances in the testing sets).
m
SSN
Primal-Dual
Inexact Uzawa
POD
iUzawa-Net-F
iUzawa-Net-S
64
3.64
209.23
29.06
3.37
6
10
128
2.86
214.19
21.61
3.53
6
10
256
3.08
215.75
22.02
3.69
6
10
Table 5.8: Number of iterations or layers of the SSN method, the primal-dual method, the inexact Uzawa method, the POD method, the iUzawa-Net-F , and the iUzawa-Net-S for solving ( 5.3 ) under different resolutions m (averaged over instances in the testing sets).
Method
m=mT
εrel(u^)
εabs(u^)
Mean
SD
Mean
SD
iUzawa-Net-F
32
3.43×10−3
1.68×10−3
7.19×10−3
2.03×10−3
64
1.45×10−2
3.59×10−3
3.20×10−2
1.03×10−2
128
2.00×10−2
3.88×10−3
4.74×10−2
1.46×10−2
256
2.19×10−2
3.56×10−3
5.96×10−2
2.21×10−2
iUzawa-Net-S
32
4.96×10−3
2.83×10−3
1.02×10−2
3.17×10−3
Table 5.9: Numerical results of iUzawa-Net-F and iUzawa-Net-S for solving ( 5.4 ).
Figure 5.3 : Computed optimal control for a single instance of ( 5.4 ) with m=mT=32 .
m=mT
SSN
Primal-Dual
Inexact Uzawa
POD
iUzawa-Net-F
iUzawa-Net-S
32
3.27×103
355.4
180.0
16.0
0.7
0.7
64
7.81×103
2.08×103
457.6
95.0
1.6
1.6
128
3.63×104
8.75×103
2.69×103
722.8
7.1
7.1
256
2.79×105
7.79×104
2.85×104
5.70×103
50.6
50.8
Table 5.10: Computation time (milliseconds) of the SSN method, the primal-dual method, the inexact Uzawa method, the POD method, the iUzawa-Net-F , and the iUzawa-Net-S for solving ( 5.4 ) with different resolutions m and mT (averaged over instances in the testing sets).
We develop deep learning-based approximation methods for fully nonlinear second-order PDEs on separable Hilbert spaces, such as HJB equations for infinite-dimensional control, by parameterizing solutions via Hilbert--Galerkin Neural Operators (HGNOs). We prove the first Universal Approximation Theorems (UATs) which are sufficiently powerful to address these problems, based on novel topologies for Hessian terms and corresponding novel continuity assumptions on the fully nonlinear operator. These topologies are non-sequential and non-metrizable, making the problem delicate. In particular, we prove UATs for functions on Hilbert spaces, together with their Fréchet derivatives up to second order, and for unbounded operators applied to the first derivative, ensuring that HGNOs are able to approximate all the PDE terms. For control problems, we further prove UATs for optimal feedback controls in terms of our approximating value function HGNO. We develop numerical training methods, which we call Deep Hilbert--Galerkin and Hilbert Actor-Critic (reinforcement learning) Methods, for these problems by minimizing the Lμ2(H)-norm of the residual of the PDE on the whole Hilbert space, not just a projected PDE to finite dimensions. This is the first paper to propose such an approach. The models considered arise in many applied sciences, such as functional differential equations in physics and Kolmogorov and HJB PDEs related to controlled PDEs, SPDEs, path-dependent systems, partially observed stochastic systems, and mean-field SDEs. We numerically solve examples of Kolmogorov and HJB PDEs related to the optimal control of deterministic and stochastic heat and Burgers' equations, demonstrating the promise of our deep learning-based approach.
Samuel N. Cohen, Filippo de Feo, Jackson Hebner +1
Mathematical Institute, University of Oxford · Institut für Mathematik, Technische Universität Berlin
We study physics-informed neural networks (PINNs) as numerical tools for the optimal control of semilinear partial differential equations. We first recall the classical direct and indirect viewpoints for optimal control of PDEs, and then present two PINN formulations: a direct formulation based on minimizing the objective under the state constraint, and an indirect formulation based on the first-order optimality system. For a class of semilinear parabolic equations, we derive the state equation, the adjoint equation, and the stationarity condition in a form consistent with continuous-time Pontryagin-type optimality conditions. We then specialize the framework to an Allen-Cahn control problem and compare three numerical approaches: (i) a discretize-then-optimize adjoint method, (ii) a direct PINN, and (iii) an indirect PINN. Numerical results show that the PINN parameterization has an implicit regularizing effect, in the sense that it tends to produce smoother control profiles. They also indicate that the indirect PINN more faithfully preserves the PDE contraint and optimality structure and yields a more accurate neural approximation than the direct PINN.
Zhen Zhang, Shanqing Liu, Alessandro Alla +2
Division of Applied Mathematics, Brown University · Department of Mathematics at Sapienza, Università di Roma
This paper presents HUANet, a constrained deep neural network architecture that unrolls the Alternating Direction Method of Multipliers (ADMM) into a trainable neural network for accelerating parametric constrained convex optimization. Existing end-to-end learning methods operate as black-box mappings from parameters to solutions, often without explicitly incorporating optimality principles or guaranteeing constraint satisfaction. To address these limitations, HUANet embeds a hard-constrained neural network within each unrolled ADMM iteration, where a differentiable correction stage enforces the affine equalities of the primal subproblem. Furthermore, we incorporate first-order optimality conditions into a self-supervised training loss to promote the convergence of the proposed unrolled algorithm. Extensive numerical experiments for benchmark optimization problems and a control application demonstrate and validate the effectiveness of HUANet in accelerating constrained convex optimization solving.
Trinh Tran, Binh Nguyen, Truong X. Nghiem
Department of Electrical and Computer Engineering, University of Central Florida, Orlando, FL 32816, USA