cs.LGOct 5, 2026

Inferring physical fields in coupled systems with unknown parameters from incomplete observations using physics-constrained attentive neural operators

Authors: Shilun Wei, Xiaoqiang Sun, Wei Li, Kejun Tang

Organizations: School of Mathematics, Sun Yat-sen University, Guangzhou, China · School of Sciences, Great Bay University, Dongguan, China · John Hopcroft Center for Computer Science, School of Computer Science, Shanghai Jiao Tong University, Shanghai, China

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Abstract

Given incomplete measurements of a single physical field in a coupled system with unknown parameters, can we infer its full physical state and identify the underlying parameters? This problem is challenging because multiple coupled fields must be reconstructed simultaneously from limited observations of only one, while the system parameters are unknown. In this work, we propose a machine learning framework for full-field reconstruction and parameter identification of unknown physical systems from sparse observations of a single physical field. Specifically, the cross-attention encoder propagates sparse sensor observations onto a regular grid to construct a sensor-conditioned latent representation, while a Fourier neural operator (FNO) decoder captures global spatial dependencies to reconstruct all coupled physical fields. The network parameters and unknown physical parameters are jointly optimized by minimizing observation losses, governing equation residuals, and boundary/initial condition constraints. The proposed approach is validated on two- and three-dimensional lid-driven cavity flows, a two-dimensional cylinder wake, and a two-dimensional non-ideal magnetohydrodynamics problem, demonstrating the recovery performance of unobserved fields and physical parameters from incomplete observations.

Figures & tables

Figure 1 : Overall architecture of the proposed physics-constrained attentive FNO framework.
BenchmarkSparse observationsInferred fields and parameters
2D lid-driven cavityBoundary pressure p∣∂Ωp|_{\partial\Omega} . Ns∈{100,200,300}N_{s}\in\{100,200,300\} at Re=3200Re=3200 ; Ns∈{200,300,400}N_{s}\in\{200,300,400\} at Re=5000Re=5000 .Full field (u,v,p)(u,v,p) and ReRe .
3D lid-driven cavityBoundary pressure p∣∂Ωp|_{\partial\Omega} ; Ns∈{500,1000,1500}N_{s}\in\{500,1000,1500\} .Full field (u,v,w,p)(u,v,w,p) and ReRe .
2D cylinder wakeBoundary velocity (u,v)∣∂Ω×(0,T](u,v)|_{\partial\Omega\times(0,T]} ; Ns∈{13,000,14,000,15,000}N_{s}\in\{13{,}000,14{,}000,15{,}000\} . No pressure observations.Time-dependent field (u,v,p)(x,y,t)(u,v,p)(x,y,t) , ReRe , and λ1\lambda_{1} .
2D non-ideal MHDMagnetic field (Bx,By)(B_{x},B_{y}) ; NB∈{5,000,10,000,15,000}N_{B}\in\{5{,}000,10{,}000,15{,}000\} . Optional pressure observations: none, boundary, or interior; Np=10,000N_{p}=10{,}000 when used.Space–time fields (ρ,vx,vy,Bx,By,p)(\rho,v_{x},v_{y},B_{x},B_{y},p) , current density jzj_{z} , and parameters ν,η\nu,\eta .
Table 1 : Inverse-problem settings for the four benchmarks.
# SensorsNoise levele∣u∣e_{|\mathbf{u}|}eue_{u}eve_{v}epe_{p}eRee_{\mathrm{Re}}
1000.007.492×10−27.492\times 10^{-2}6.932×10−26.932\times 10^{-2}8.427×10−28.427\times 10^{-2}2.409×10−12.409\times 10^{-1}7.967×10−27.967\times 10^{-2}
2000.004.854×10−24.854\times 10^{-2}4.947×10−24.947\times 10^{-2}5.316×10−25.316\times 10^{-2}1.078×10−11.078\times 10^{-1}6.269×10−26.269\times 10^{-2}
3000.003.846×10−23.846\times 10^{-2}3.563×10−23.563\times 10^{-2}4.524×10−24.524\times 10^{-2}7.701×10−27.701\times 10^{-2}9.220×10−29.220\times 10^{-2}
1000.014.638×10−24.638\times 10^{-2}4.801×10−24.801\times 10^{-2}5.066×10−25.066\times 10^{-2}1.027×10−11.027\times 10^{-1}7.105×10−27.105\times 10^{-2}
2000.014.941×10−24.941\times 10^{-2}5.093×10−25.093\times 10^{-2}5.364×10−25.364\times 10^{-2}1.051×10−11.051\times 10^{-1}8.565×10−28.565\times 10^{-2}
3000.014.064×10−24.064\times 10^{-2}3.746×10−23.746\times 10^{-2}4.779×10−24.779\times 10^{-2}8.460×10−28.460\times 10^{-2}1.105×10−11.105\times 10^{-1}
Table 2 : Reconstruction and parameter inversion errors for the 2D lid-driven cavity inverse problem at Re=3200\mathrm{Re}=3200 under different numbers of pressure observations and noise levels.
Re\mathrm{Re}Methode∣u∣e_{|\mathbf{u}|}eue_{u}eve_{v}epe_{p}eRee_{\mathrm{Re}}
32003200Vanilla PINN7.712×10−17.712\times 10^{-1}8.271×10−18.271\times 10^{-1}9.387×10−19.387\times 10^{-1}7.537×10−17.537\times 10^{-1}2.155×10−12.155\times 10^{-1}
Ours3.846×10−2\mathbf{3.846\times 10^{-2}}3.563×10−2\mathbf{3.563\times 10^{-2}}4.524×10−2\mathbf{4.524\times 10^{-2}}7.701×10−2\mathbf{7.701\times 10^{-2}}9.220×10−2\mathbf{9.220\times 10^{-2}}
50005000Vanilla PINN7.451×10−17.451\times 10^{-1}7.778×10−17.778\times 10^{-1}9.309×10−19.309\times 10^{-1}7.161×10−17.161\times 10^{-1}4.290×10−14.290\times 10^{-1}
Ours1.030×10−1\mathbf{1.030\times 10^{-1}}9.743×10−2\mathbf{9.743\times 10^{-2}}1.197×10−1\mathbf{1.197\times 10^{-1}}2.959×10−1\mathbf{2.959\times 10^{-1}}5.999×10−2\mathbf{5.999\times 10^{-2}}
Table 3 : Comparison with vanilla PINN for the 2D lid-driven cavity inverse problem using sparse boundary pressure observations without noise. The numbers of observations are 300 and 400 for Re=3200\mathrm{Re}=3200 and 50005000 , respectively.
CaseFeedback layersSensor weightinge∣u∣e_{|\mathbf{u}|}eue_{u}eve_{v}epe_{p}eRee_{\mathrm{Re}}
Full model2Residual4.854×10−24.854\times 10^{-2}4.947×10−24.947\times 10^{-2}5.317×10−25.317\times 10^{-2}1.078×10−11.078\times 10^{-1}6.269×10−26.269\times 10^{-2}
No sensor weighting2None5.312×10−25.312\times 10^{-2}5.495×10−25.495\times 10^{-2}5.677×10−25.677\times 10^{-2}1.093×10−11.093\times 10^{-1}1.137×10−11.137\times 10^{-1}
One feedback layer1Residual5.217×10−15.217\times 10^{-1}5.678×10−15.678\times 10^{-1}5.744×10−15.744\times 10^{-1}4.536×10−14.536\times 10^{-1}1.055×10−21.055\times 10^{-2}
One feedback layer, no sensor weighting1None5.564×10−15.564\times 10^{-1}6.188×10−16.188\times 10^{-1}6.434×10−16.434\times 10^{-1}5.089×10−15.089\times 10^{-1}2.799×10−22.799\times 10^{-2}
Table 4 : Ablation study for the 2D lid-driven cavity inverse problem at Re=3200\mathrm{Re}=3200 with 200 pressure observations and no observation noise.
Figure 2 : Predicted fields and point-wise errors for the 2D lid-driven cavity inverse problem at Re=3200\mathrm{Re}=3200 . The columns show uu , vv , ∣u∣|\mathbf{u}| , and pp . The first two rows give the noise-free case. The last two rows give the case with 1%1\% pressure noise.
Figure 3 : Predicted fields and point-wise errors for the 2D lid-driven cavity inverse problem at Re=5000\mathrm{Re}=5000 . The columns show uu , vv , ∣u∣|\mathbf{u}| , and pp . The first two rows give the noise-free case. The last two rows give the case with 1%1\% pressure noise.
# SensorsNoise levele∣u∣e_{|\mathbf{u}|}eue_{u}eve_{v}epe_{p}eRee_{\mathrm{Re}}
2000.006.072×10−16.072\times 10^{-1}7.594×10−17.594\times 10^{-1}8.456×10−18.456\times 10^{-1}7.072×10−17.072\times 10^{-1}1.081×10−11.081\times 10^{-1}
3000.001.502×10−11.502\times 10^{-1}1.408×10−11.408\times 10^{-1}1.655×10−11.655\times 10^{-1}2.412×10−12.412\times 10^{-1}3.688×10−13.688\times 10^{-1}
4000.001.030×10−11.030\times 10^{-1}9.743×10−29.743\times 10^{-2}1.197×10−11.197\times 10^{-1}2.959×10−12.959\times 10^{-1}5.999×10−25.999\times 10^{-2}
2000.016.037×10−16.037\times 10^{-1}7.796×10−17.796\times 10^{-1}8.824×10−18.824\times 10^{-1}7.437×10−17.437\times 10^{-1}2.962×10−32.962\times 10^{-3}
3000.011.451×10−11.451\times 10^{-1}1.364×10−11.364\times 10^{-1}1.596×10−11.596\times 10^{-1}2.337×10−12.337\times 10^{-1}3.531×10−13.531\times 10^{-1}
4000.011.436×10−11.436\times 10^{-1}1.374×10−11.374\times 10^{-1}1.590×10−11.590\times 10^{-1}2.906×10−12.906\times 10^{-1}4.181×10−24.181\times 10^{-2}
Table 5 : Reconstruction and parameter inversion errors for the 2D lid-driven cavity inverse problem at Re=5000\mathrm{Re}=5000 under different numbers of pressure observations and noise levels.
CaseFeedback layersSensor weightinge∣u∣e_{|\mathbf{u}|}eue_{u}eve_{v}epe_{p}eRee_{\mathrm{Re}}
Full model2Residual1.030×10−11.030\times 10^{-1}9.743×10−29.743\times 10^{-2}1.197×10−11.197\times 10^{-1}2.959×10−12.959\times 10^{-1}5.999×10−25.999\times 10^{-2}
No sensor weighting2None5.380×10−15.380\times 10^{-1}7.385×10−17.385\times 10^{-1}8.496×10−18.496\times 10^{-1}7.356×10−17.356\times 10^{-1}2.088×10−12.088\times 10^{-1}
One feedback layer1Residual5.548×10−15.548\times 10^{-1}8.461×10−18.461\times 10^{-1}1.058×1001.058\times 10^{0}8.056×10−18.056\times 10^{-1}1.061×1001.061\times 10^{0}
One feedback layer, no sensor weighting1None5.608×10−15.608\times 10^{-1}8.719×10−18.719\times 10^{-1}1.117×1001.117\times 10^{0}9.838×10−19.838\times 10^{-1}8.701×10−18.701\times 10^{-1}
Table 6 : Ablation study for the 2D lid-driven cavity inverse problem at Re=5000\mathrm{Re}=5000 with 400 pressure observations and no observation noise.
Figure 4 : Relative L2L_{2} errors for the two-dimensional lid-driven cavity inverse problem. The first row reports the results at Re=3200\mathrm{Re}=3200 , and the second row reports the results at Re=5000\mathrm{Re}=5000 . The columns correspond to the errors of uu , vv , and pp , respectively.
Figure 5 : Histories of the identified Reynolds number for the two-dimensional lid-driven cavity inverse problem. The horizontal dashed lines indicate the corresponding ground-truth values.
Noise level# Pressure obs.e∣u∣e_{|\mathbf{u}|}eue_{u}eve_{v}ewe_{w}epe_{p}eRee_{\mathrm{Re}}
0.005001.086×10−11.086\times 10^{-1}1.211×10−11.211\times 10^{-1}9.763×10−29.763\times 10^{-2}1.048×10−11.048\times 10^{-1}1.489×10−11.489\times 10^{-1}1.471×10−21.471\times 10^{-2}
10001.034×10−11.034\times 10^{-1}1.157×10−11.157\times 10^{-1}9.309×10−29.309\times 10^{-2}8.912×10−28.912\times 10^{-2}1.311×10−11.311\times 10^{-1}6.479×10−26.479\times 10^{-2}
15001.019×10−11.019\times 10^{-1}1.142×10−11.142\times 10^{-1}9.209×10−29.209\times 10^{-2}7.017×10−27.017\times 10^{-2}1.304×10−11.304\times 10^{-1}7.833×10−27.833\times 10^{-2}
0.015001.073×10−11.073\times 10^{-1}1.197×10−11.197\times 10^{-1}9.591×10−29.591\times 10^{-2}1.113×10−11.113\times 10^{-1}1.444×10−11.444\times 10^{-1}2.912×10−22.912\times 10^{-2}
10001.025×10−11.025\times 10^{-1}1.149×10−11.149\times 10^{-1}9.176×10−29.176\times 10^{-2}8.767×10−28.767\times 10^{-2}1.296×10−11.296\times 10^{-1}7.311×10−27.311\times 10^{-2}
15001.040×10−11.040\times 10^{-1}1.158×10−11.158\times 10^{-1}9.534×10−29.534\times 10^{-2}7.447×10−27.447\times 10^{-2}1.317×10−11.317\times 10^{-1}6.282×10−26.282\times 10^{-2}
Table 7 : Reconstruction and parameter inversion errors for the 3D lid-driven cavity inverse problem under different numbers of boundary pressure observations and noise levels.
CaseFeedback layersSensor weightinge∣u∣e_{|\mathbf{u}|}eue_{u}eve_{v}ewe_{w}epe_{p}eRee_{\mathrm{Re}}
Full model1Residual1.019×10−11.019\times 10^{-1}1.142×10−11.142\times 10^{-1}9.209×10−29.209\times 10^{-2}7.017×10−27.017\times 10^{-2}1.304×10−11.304\times 10^{-1}7.833×10−27.833\times 10^{-2}
No sensor weighting1None1.022×10−11.022\times 10^{-1}1.145×10−11.145\times 10^{-1}9.223×10−29.223\times 10^{-2}7.005×10−27.005\times 10^{-2}1.325×10−11.325\times 10^{-1}7.253×10−27.253\times 10^{-2}
No feedback layer0Residual9.955×10−29.955\times 10^{-2}1.126×10−11.126\times 10^{-1}8.810×10−28.810\times 10^{-2}6.890×10−26.890\times 10^{-2}1.312×10−11.312\times 10^{-1}9.605×10−29.605\times 10^{-2}
No feedback layer, no sensor weighting0None9.971×10−29.971\times 10^{-2}1.128×10−11.128\times 10^{-1}8.817×10−28.817\times 10^{-2}6.791×10−26.791\times 10^{-2}1.324×10−11.324\times 10^{-1}9.226×10−29.226\times 10^{-2}
Table 8 : Ablation study for the 3D lid-driven cavity inverse problem at Re=500\mathrm{Re}=500 with 1500 boundary pressure observations and no observation noise.
Methode∣u∣e_{|\mathbf{u}|}eue_{u}eve_{v}ewe_{w}epe_{p}eRee_{\mathrm{Re}}
Vanilla PINN5.851×10−15.851\times 10^{-1}5.947×10−15.947\times 10^{-1}9.285×10−19.285\times 10^{-1}2.490×1002.490\times 10^{0}4.789×10−14.789\times 10^{-1}2.862×10−12.862\times 10^{-1}
Ours1.019×10−1\mathbf{1.019\times 10^{-1}}1.142×10−1\mathbf{1.142\times 10^{-1}}9.209×10−2\mathbf{9.209\times 10^{-2}}7.017×10−2\mathbf{7.017\times 10^{-2}}1.304×10−1\mathbf{1.304\times 10^{-1}}7.833×10−2\mathbf{7.833\times 10^{-2}}
Table 9 : Comparison with vanilla PINN for the 3D lid-driven cavity inverse problem at Re=500\mathrm{Re}=500 using 1500 boundary pressure observations without noise.
Figure 6 : Predicted fields and absolute errors for the 3D lid-driven cavity inverse problem at Re=500\mathrm{Re}=500 . The plots are shown on the mid-plane slice. The columns show uu , vv , ww , ∣u∣|\mathbf{u}| , and pp . The first two rows give the noise-free case, and the last two rows show the case with 1%1\% pressure noise.
Figure 7 : Training histories for the 3D lid-driven cavity inverse problem at Re=500\mathrm{Re}=500 . The first three panels compare the relative L2L_{2} errors of uu , vv , and the mean-free pressure pp between Attn+FNO and Vanilla PINN. The last panel shows their identified Reynolds numbers.
Noise level# Velocity obs.e∣u∣e_{|\mathbf{u}|}eue_{u}eve_{v}epe_{p}eRee_{\mathrm{Re}}eλ1e_{\lambda_{1}}
0.0013,0003.146×10−23.146\times 10^{-2}2.637×10−22.637\times 10^{-2}1.156×10−11.156\times 10^{-1}3.386×10−13.386\times 10^{-1}3.130×10−13.130\times 10^{-1}5.907×10−35.907\times 10^{-3}
14,0002.677×10−22.677\times 10^{-2}2.245×10−22.245\times 10^{-2}9.804×10−29.804\times 10^{-2}2.329×10−12.329\times 10^{-1}2.771×10−12.771\times 10^{-1}5.517×10−35.517\times 10^{-3}
15,0005.736×10−35.736\times 10^{-3}5.498×10−35.498\times 10^{-3}1.871×10−21.871\times 10^{-2}1.136×10−11.136\times 10^{-1}4.917×10−24.917\times 10^{-2}7.368×10−47.368\times 10^{-4}
0.0113,0005.612×10−25.612\times 10^{-2}4.754×10−24.754\times 10^{-2}2.086×10−12.086\times 10^{-1}3.787×10−13.787\times 10^{-1}4.477×10−14.477\times 10^{-1}3.417×10−33.417\times 10^{-3}
14,0001.580×10−21.580\times 10^{-2}1.360×10−21.360\times 10^{-2}5.672×10−25.672\times 10^{-2}2.448×10−12.448\times 10^{-1}1.459×10−11.459\times 10^{-1}4.799×10−44.799\times 10^{-4}
15,0003.427×10−23.427\times 10^{-2}2.885×10−22.885\times 10^{-2}1.260×10−11.260\times 10^{-1}2.122×10−12.122\times 10^{-1}3.176×10−13.176\times 10^{-1}3.574×10−33.574\times 10^{-3}
Table 10 : Reconstruction and parameter inversion errors for the 2D cylinder wake inverse problem under different numbers of velocity boundary observations and noise levels.
Methode∣u∣e_{|\mathbf{u}|}eue_{u}eve_{v}epe_{p}eRee_{\mathrm{Re}}eλ1e_{\lambda_{1}}
Vanilla PINN8.630×10−28.630\times 10^{-2}7.435×10−27.435\times 10^{-2}2.773×10−12.773\times 10^{-1}4.692×10−14.692\times 10^{-1}2.784×10−12.784\times 10^{-1}3.202×10−23.202\times 10^{-2}
Ours5.736×10−3\mathbf{5.736\times 10^{-3}}5.498×10−3\mathbf{5.498\times 10^{-3}}1.871×10−2\mathbf{1.871\times 10^{-2}}1.136×10−1\mathbf{1.136\times 10^{-1}}4.917×10−2\mathbf{4.917\times 10^{-2}}7.368×10−4\mathbf{7.368\times 10^{-4}}
Table 11 : Comparison with vanilla PINN for the 2D cylinder wake inverse problem using 15000 velocity boundary observations without noise.
CaseFeedback layersSensor weightinge∣u∣e_{|\mathbf{u}|}eue_{u}eve_{v}epe_{p}eRee_{\mathrm{Re}}
Full model1Residual5.736×10−35.736\times 10^{-3}5.498×10−35.498\times 10^{-3}1.871×10−21.871\times 10^{-2}1.136×10−11.136\times 10^{-1}4.917×10−24.917\times 10^{-2}
No sensor weighting1None4.360×10−24.360\times 10^{-2}3.643×10−23.643\times 10^{-2}1.608×10−11.608\times 10^{-1}3.644×10−13.644\times 10^{-1}3.994×10−13.994\times 10^{-1}
No feedback0Residual3.823×10−23.823\times 10^{-2}3.204×10−23.204\times 10^{-2}1.405×10−11.405\times 10^{-1}2.271×10−12.271\times 10^{-1}3.561×10−13.561\times 10^{-1}
No feedback, no sensor weighting0None6.422×10−26.422\times 10^{-2}5.326×10−25.326\times 10^{-2}2.381×10−12.381\times 10^{-1}3.725×10−13.725\times 10^{-1}5.005×10−15.005\times 10^{-1}
Table 12 : Ablation study for the 2D cylinder wake inverse problem with 15000 velocity boundary observations and no observation noise.
Figure 8 : Predicted fields and absolute errors for the 2D cylinder wake inverse problem with clean boundary velocity data. The columns show ∣u∣|\mathbf{u}| , uu , vv , and mean-free pressure pp . The first two rows give the results at t=10t=10 . The last two rows give the results at t=19t=19 .
Figure 9 : Predicted fields and absolute errors for the 2D cylinder wake inverse problem with 1%1\% noisy boundary velocity data. The columns show ∣u∣|\mathbf{u}| , uu , vv , and mean-free pressure pp . The first two rows give the results at t=10t=10 . The last two rows give the results at t=19t=19 .
Figure 10 : Training histories for the 2D cylinder wake inverse problem. The first three panels show the relative L2L_{2} errors of uu , vv , and the pressure pp , while the last panel shows the identified Reynolds number. The compared methods are Attn+FNO, MeanPool+FNO, MLP+FNO, and Voronoi+FNO.
Noise levelPressure obs.NbN_{b}NpN_{p}eue_{u}eve_{v}ebxe_{b_{x}}ebye_{b_{y}}eρe_{\rho}epe_{p}eνe_{\nu}eηe_{\eta}
0.00Internal5000100001.205×10−11.205\times 10^{-1}1.213×10−11.213\times 10^{-1}5.354×10−25.354\times 10^{-2}5.549×10−25.549\times 10^{-2}7.475×10−27.475\times 10^{-2}1.767×10−11.767\times 10^{-1}7.786×10−27.786\times 10^{-2}8.519×10−28.519\times 10^{-2}
Internal10000100001.166×10−11.166\times 10^{-1}1.153×10−11.153\times 10^{-1}4.535×10−24.535\times 10^{-2}4.764×10−24.764\times 10^{-2}7.225×10−27.225\times 10^{-2}1.750×10−11.750\times 10^{-1}8.348×10−28.348\times 10^{-2}8.668×10−28.668\times 10^{-2}
None15,000–2.349×10−12.349\times 10^{-1}2.468×10−12.468\times 10^{-1}4.762×10−24.762\times 10^{-2}5.226×10−25.226\times 10^{-2}1.051×10−11.051\times 10^{-1}4.448×10−14.448\times 10^{-1}5.560×10−25.560\times 10^{-2}7.423×10−27.423\times 10^{-2}
boundary15,000100001.694×10−11.694\times 10^{-1}1.708×10−11.708\times 10^{-1}4.503×10−24.503\times 10^{-2}4.879×10−24.879\times 10^{-2}9.099×10−29.099\times 10^{-2}2.850×10−12.850\times 10^{-1}3.121×10−23.121\times 10^{-2}5.600×10−25.600\times 10^{-2}
Internal15,000100001.197×10−11.197\times 10^{-1}1.206×10−11.206\times 10^{-1}4.017×10−24.017\times 10^{-2}4.286×10−24.286\times 10^{-2}7.023×10−27.023\times 10^{-2}1.779×10−11.779\times 10^{-1}1.425×10−11.425\times 10^{-1}1.358×10−11.358\times 10^{-1}
0.01Internal5000100001.197×10−11.197\times 10^{-1}1.202×10−11.202\times 10^{-1}5.408×10−25.408\times 10^{-2}5.655×10−25.655\times 10^{-2}7.502×10−27.502\times 10^{-2}1.777×10−11.777\times 10^{-1}7.838×10−27.838\times 10^{-2}8.610×10−28.610\times 10^{-2}
Table 13 : Reconstruction and parameter inversion errors for the non-ideal MHD inverse problem under different magnetic-field observations, pressure-observation settings, and noise levels. Here NbN_{b} and NpN_{p} denote the numbers of magnetic-field and pressure observations. “None”, “boundary”, and “Internal” denote no pressure observation, boundary pressure observation, and internal pressure observation.
CaseFeedback layersSensor weightingeue_{u}eve_{v}ebxe_{b_{x}}ebye_{b_{y}}eρe_{\rho}epe_{p}eνe_{\nu}eηe_{\eta}
Full model2Residual1.197×10−11.197\times 10^{-1}1.206×10−11.206\times 10^{-1}4.017×10−24.017\times 10^{-2}4.286×10−24.286\times 10^{-2}7.023×10−27.023\times 10^{-2}1.779×10−11.779\times 10^{-1}1.425×10−11.425\times 10^{-1}1.358×10−11.358\times 10^{-1}
No sensor weighting2None1.199×10−11.199\times 10^{-1}1.215×10−11.215\times 10^{-1}4.012×10−24.012\times 10^{-2}4.351×10−24.351\times 10^{-2}7.208×10−27.208\times 10^{-2}1.781×10−11.781\times 10^{-1}8.176×10−28.176\times 10^{-2}8.787×10−28.787\times 10^{-2}
No feedback layer1Residual1.154×10−11.154\times 10^{-1}1.174×10−11.174\times 10^{-1}4.115×10−24.115\times 10^{-2}4.385×10−24.385\times 10^{-2}7.057×10−27.057\times 10^{-2}1.717×10−11.717\times 10^{-1}1.425×10−11.425\times 10^{-1}1.358×10−11.358\times 10^{-1}
No feedback layer, no sensor weighting1None1.214×10−11.214\times 10^{-1}1.218×10−11.218\times 10^{-1}4.240×10−24.240\times 10^{-2}4.482×10−24.482\times 10^{-2}7.051×10−27.051\times 10^{-2}1.812×10−11.812\times 10^{-1}4.927×10−14.927\times 10^{-1}4.947×10−14.947\times 10^{-1}
Table 14 : Ablation study for the 2D non-ideal MHD inverse problem with 15000 internal magnetic-field observations and 10000 internal pressure observations.
Methodeue_{u}eve_{v}ebxe_{b_{x}}ebye_{b_{y}}eρe_{\rho}epe_{p}eνe_{\nu}eηe_{\eta}
Vanilla PINN3.378×10−13.378\times 10^{-1}3.142×10−13.142\times 10^{-1}8.348×10−28.348\times 10^{-2}9.135×10−29.135\times 10^{-2}4.392×10−14.392\times 10^{-1}7.149×10−17.149\times 10^{-1}5.244×10−15.244\times 10^{-1}1.527×10−11.527\times 10^{-1}
Ours2.349×10−1\mathbf{2.349\times 10^{-1}}2.468×10−1\mathbf{2.468\times 10^{-1}}4.762×10−2\mathbf{4.762\times 10^{-2}}5.226×10−2\mathbf{5.226\times 10^{-2}}1.051×10−1\mathbf{1.051\times 10^{-1}}4.448×10−1\mathbf{4.448\times 10^{-1}}5.560×10−2\mathbf{5.560\times 10^{-2}}7.423×10−2\mathbf{7.423\times 10^{-2}}
Table 15 : Comparison with vanilla PINN for the non-ideal MHD inverse problem using 15000 magnetic-field observation locations without pressure observations or noise.
Figure 11 : Predicted fields and absolute errors for the 2D non-ideal MHD inverse problem with clean magnetic-field and internal pressure data. The columns show uu , vv , BxB_{x} , ByB_{y} , mean-free pressure pp , and density ρ\rho . The row pairs correspond to frames 50, 200, and 399.
Figure 12 : Predicted fields and absolute errors for the 2D non-ideal MHD inverse problem with 1%1\% noisy magnetic-field and internal pressure data. The columns show uu , vv , BxB_{x} , ByB_{y} , mean-free pressure pp , and density ρ\rho . The row pairs correspond to frames 50, 200, and 399.
Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

# SensorsNoise levele∣u∣e_{|\mathbf{u}|}eue_{u}eve_{v}epe_{p}eRee_{\mathrm{Re}}
1000.00(8.037±2.187)×10−2(8.037\pm 2.187)\times 10^{-2}(7.558±2.184)×10−2(7.558\pm 2.184)\times 10^{-2}(8.948±2.196)×10−2(8.948\pm 2.196)\times 10^{-2}(2.598±0.193)×10−1(2.598\pm 0.193)\times 10^{-1}(1.239±0.557)×10−1(1.239\pm 0.557)\times 10^{-1}
2000.00(5.136±0.384)×10−2(5.136\pm 0.384)\times 10^{-2}(5.197±0.388)×10−2(5.197\pm 0.388)\times 10^{-2}(5.669±0.360)×10−2(5.669\pm 0.360)\times 10^{-2}(1.479±0.794)×10−1(1.479\pm 0.794)\times 10^{-1}(6.263±3.650)×10−2(6.263\pm 3.650)\times 10^{-2}
3000.00(4.799±1.747)×10−2(4.799\pm 1.747)\times 10^{-2}(4.487±1.774)×10−2(4.487\pm 1.774)\times 10^{-2}(5.607±1.946)×10−2(5.607\pm 1.946)\times 10^{-2}(1.145±0.682)×10−1(1.145\pm 0.682)\times 10^{-1}(1.444±0.823)×10−1(1.444\pm 0.823)\times 10^{-1}
1000.01(5.718±1.010)×10−2(5.718\pm 1.010)\times 10^{-2}(5.379±0.834)×10−2(5.379\pm 0.834)\times 10^{-2}(6.511±1.173)×10−2(6.511\pm 1.173)\times 10^{-2}(2.176±0.685)×10−1(2.176\pm 0.685)\times 10^{-1}(9.281±4.544)×10−2(9.281\pm 4.544)\times 10^{-2}
2000.01(5.017±0.244)×10−2(5.017\pm 0.244)\times 10^{-2}(5.138±0.227)×10−2(5.138\pm 0.227)\times 10^{-2}(5.452±0.237)×10−2(5.452\pm 0.237)\times 10^{-2}(1.083±0.034)×10−1(1.083\pm 0.034)\times 10^{-1}(7.717±1.085)×10−2(7.717\pm 1.085)\times 10^{-2}
3000.01(4.120±0.165)×10−2(4.120\pm 0.165)\times 10^{-2}(3.806±0.146)×10−2(3.806\pm 0.146)\times 10^{-2}(4.830±0.189)×10−2(4.830\pm 0.189)\times 10^{-2}(9.036±0.944)×10−2(9.036\pm 0.944)\times 10^{-2}(1.155±0.081)×10−1(1.155\pm 0.081)\times 10^{-1}
Appendix
Table 16 : Random-seed evaluation for the 2D lid-driven cavity inverse problem at Re=3200\mathrm{Re}=3200 . All settings are reported as mean ±\pm standard deviation over five runs.
NsN_{s}σ^min⁡(0.005)\widehat{\sigma}_{\min}(0.005)σ^min⁡(0.010)\widehat{\sigma}_{\min}(0.010)σ^min⁡(0.020)\widehat{\sigma}_{\min}(0.020)σ^min⁡rms(0.010)\widehat{\sigma}_{\min}^{\rm rms}(0.010)Δτ\Delta_{\tau}
2000.4009470.4069440.4099010.0287751.47%1.47\%
3000.3743310.3820870.3859470.0220602.03%2.03\%
4000.4671870.4761200.4805510.0238061.88%1.88\%
Appendix
Table 17 : Finite-difference diagnostic for Assumption A2 in the two-dimensional cavity problem at Re=5000\mathrm{Re}=5000 .

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