math.OCFeb 12, 2026

Learning to Control: The iUzawa-Net for Nonsmooth Optimal Control of Linear PDEs

Authors: Yongcun Song, Xiaoming Yuan, Hangrui Yue, Tianyou Zeng

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.

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Abstract

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\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.

Figures & tables

Figure 2.1 : The neural network architecture of QSk\mathcal{Q}_{S}^{k} .
Figure 2.2 : The neural network architecture of QAk\mathcal{Q}_{A}^{k} .
Figure 2.3 : An overview of the iUzawa-Net ( 1.7 ).
mm#\# of active ui∗u_{i}^{*}∣A(ui∗)∣/∣Ω∣|A(u_{i}^{*})|/|\Omega| for active ui∗u_{i}^{*}
TrainingTestingTraining MeanTesting Mean
64641145011450142214220.21160.21160.21280.2128
128128—14191419—0.21300.2130
256256—14331433—0.22560.2256
Table 5.1: Statistics of active optimal control and active sets in the training and testing datasets of different resolutions mm . A dash (—) indicates that the corresponding data is not applicable.
Methodmmεrel(u^)\varepsilon_{\mathrm{rel}}(\hat{u})εabs(u^)\varepsilon_{\mathrm{abs}}(\hat{u})
MeanSDMeanSD
iUzawa-Net-F64642.00×10−32.00\times 10^{-3}0.97×10−30.97\times 10^{-3}4.05×10−34.05\times 10^{-3}1.85×10−31.85\times 10^{-3}
1281282.29×10−32.29\times 10^{-3}0.99×10−30.99\times 10^{-3}4.60×10−34.60\times 10^{-3}1.76×10−31.76\times 10^{-3}
2562562.38×10−32.38\times 10^{-3}0.85×10−30.85\times 10^{-3}4.99×10−34.99\times 10^{-3}2.13×10−32.13\times 10^{-3}
iUzawa-Net-S64643.67×10−33.67\times 10^{-3}3.02×10−33.02\times 10^{-3}7.03×10−37.03\times 10^{-3}3.74×10−33.74\times 10^{-3}
1281283.73×10−33.73\times 10^{-3}2.79×10−32.79\times 10^{-3}7.15×10−37.15\times 10^{-3}3.69×10−33.69\times 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=64m=64 .
mmSSNPrimal-DualInexact UzawaPODiUzawa-Net-FiUzawa-Net-S
646479.279.225.125.142.742.71.11.10.30.30.50.5
128128176.7176.7218.7218.7177.1177.13.83.80.40.40.60.6
256256773.9773.9966.0966.0355.0355.018.318.30.60.61.01.0
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 mm (averaged over instances in the testing sets).
mmSSNPrimal-DualInexact UzawaPODiUzawa-Net-FiUzawa-Net-S
64642.592.5910.7710.7717.9417.942.362.36661010
1281281.911.9110.7810.787.717.712.432.43661010
2562561.941.9410.7810.786.576.572.602.60661010
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 mm (averaged over instances in the testing sets).
iUzawa-Net-FiUzawa-Net-S
LLParametersεrel(u^)\varepsilon_{\mathrm{rel}}(\hat{u})LLParametersεrel(u^)\varepsilon_{\mathrm{rel}}(\hat{u})
MeanSDMeanSD
2290,69890{,}6986.95×10−36.95\times 10^{-3}4.53×10−34.53\times 10^{-3}2245,34945{,}3491.21×10−21.21\times 10^{-2}1.12×10−21.12\times 10^{-2}
33136,047136{,}0474.14×10−34.14\times 10^{-3}2.38×10−32.38\times 10^{-3}3345,34945{,}3496.15×10−36.15\times 10^{-3}2.95×10−32.95\times 10^{-3}
66272,094272{,}0942.00×10−32.00\times 10^{-3}0.97×10−30.97\times 10^{-3}6645,34945{,}3495.88×10−35.88\times 10^{-3}3.21×10−33.21\times 10^{-3}
1010453,490453{,}4901.63×10−31.63\times 10^{-3}0.77×10−30.77\times 10^{-3}101045,34945{,}3493.67×10−33.67\times 10^{-3}3.02×10−33.02\times 10^{-3}
Table 5.5: Relative errors εrel(u^)\varepsilon_{\mathrm{rel}}(\hat{u}) on the testing set with m=64m=64 of the iUzawa-Net-F and the iUzawa-Net-S with different numbers of layers LL for solving ( 5.1 ).
Figure 5.2 : Computed optimal control for a single instance of ( 5.3 ) with m=64m=64 .
Methodmmεrel(u^)\varepsilon_{\mathrm{rel}}(\hat{u})εabs(u^)\varepsilon_{\mathrm{abs}}(\hat{u})
MeanSDMeanSD
iUzawa-Net-F64647.14×10−37.14\times 10^{-3}1.02×10−21.02\times 10^{-2}1.85×10−21.85\times 10^{-2}1.82×10−21.82\times 10^{-2}
1281285.52×10−35.52\times 10^{-3}4.48×10−34.48\times 10^{-3}1.50×10−21.50\times 10^{-2}1.10×10−21.10\times 10^{-2}
2562566.89×10−36.89\times 10^{-3}6.68×10−36.68\times 10^{-3}1.69×10−21.69\times 10^{-2}1.36×10−21.36\times 10^{-2}
iUzawa-Net-S64644.71×10−34.71\times 10^{-3}5.40×10−35.40\times 10^{-3}1.12×10−21.12\times 10^{-2}8.81×10−38.81\times 10^{-3}
1281284.12×10−34.12\times 10^{-3}2.96×10−32.96\times 10^{-3}1.02×10−21.02\times 10^{-2}7.72×10−37.72\times 10^{-3}
Table 5.6: Numerical results for solving ( 5.3 ).
mmSSNPrimal-DualInexact UzawaPODiUzawa-Net-FiUzawa-Net-S
64644.29×1034.29\times 10^{3}1.14×1031.14\times 10^{3}171.1171.10.80.80.30.30.50.5
1281282.14×1042.14\times 10^{4}8.71×1038.71\times 10^{3}1.17×1031.17\times 10^{3}2.22.20.40.40.60.6
2562566.07×1046.07\times 10^{4}2.30×1042.30\times 10^{4}4.94×1034.94\times 10^{3}9.29.20.60.61.01.0
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 mm (averaged over instances in the testing sets).
mmSSNPrimal-DualInexact UzawaPODiUzawa-Net-FiUzawa-Net-S
64643.643.64209.23209.2329.0629.063.373.37661010
1281282.862.86214.19214.1921.6121.613.533.53661010
2562563.083.08215.75215.7522.0222.023.693.69661010
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 mm (averaged over instances in the testing sets).
Methodm=mTm=m_{T}εrel(u^)\varepsilon_{\mathrm{rel}}(\hat{u})εabs(u^)\varepsilon_{\mathrm{abs}}(\hat{u})
MeanSDMeanSD
iUzawa-Net-F32323.43×10−33.43\times 10^{-3}1.68×10−31.68\times 10^{-3}7.19×10−37.19\times 10^{-3}2.03×10−32.03\times 10^{-3}
64641.45×10−21.45\times 10^{-2}3.59×10−33.59\times 10^{-3}3.20×10−23.20\times 10^{-2}1.03×10−21.03\times 10^{-2}
1281282.00×10−22.00\times 10^{-2}3.88×10−33.88\times 10^{-3}4.74×10−24.74\times 10^{-2}1.46×10−21.46\times 10^{-2}
2562562.19×10−22.19\times 10^{-2}3.56×10−33.56\times 10^{-3}5.96×10−25.96\times 10^{-2}2.21×10−22.21\times 10^{-2}
iUzawa-Net-S32324.96×10−34.96\times 10^{-3}2.83×10−32.83\times 10^{-3}1.02×10−21.02\times 10^{-2}3.17×10−33.17\times 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=32m=m_{T}=32 .
m=mTm=m_{T}SSNPrimal-DualInexact UzawaPODiUzawa-Net-FiUzawa-Net-S
32323.27×1033.27\times 10^{3}355.4355.4180.0180.016.016.00.70.70.70.7
64647.81×1037.81\times 10^{3}2.08×1032.08\times 10^{3}457.6457.695.095.01.61.61.61.6
1281283.63×1043.63\times 10^{4}8.75×1038.75\times 10^{3}2.69×1032.69\times 10^{3}722.8722.87.17.17.17.1
2562562.79×1052.79\times 10^{5}7.79×1047.79\times 10^{4}2.85×1042.85\times 10^{4}5.70×1035.70\times 10^{3}50.650.650.850.8
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 mm and mTm_{T} (averaged over instances in the testing sets).
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