Bi-HYCO: Bi-Objective Cooperative Learning for PDE Parameter Identification under Fragmented Observations

Umberto Biccari; Jun Chen; Roberto Morales; Enrique Zuazua

cs.LGSep 6, 2026

Bi-HYCO: Bi-Objective Cooperative Learning for PDE Parameter Identification under Fragmented Observations

Authors: Umberto Biccari, Jun Chen, Roberto Morales, Enrique Zuazua

Organizations: University of Madrid · China Scholarship Council

Abstract

Physical and synthetic models may describe complementary aspects of the same PDE-governed system while receiving different, possibly fragmented, observations. We propose Bi-Objective HYCO (Bi-HYCO), a cooperative framework that retains both representations and their local observational objectives while coupling their predicted states at unlabeled interaction points. These points contain no measurements and do not augment the data; they provide a communication mechanism in the common state space. The two criteria form a vector-valued objective, and weighted scalarizations provide computational realizations. For the deterministic shared-observation algorithm with fixed interaction points, we prove sufficient decrease and finite length of the whole alternating sequence, which converges to a mixed critical point under the stated Kurdyka-Lojasiewicz-type assumptions. Elliptic transmission and two-dimensional Navier-Stokes experiments assess parameter and state reconstruction, noise and scalarization effects, and PINN/XPINN references. Ablations show that removing state interaction while retaining aggregation deteriorates parameter recovery in the tested configurations, particularly for Navier-Stokes.

Figures & tables

Figure 1. Alternating minimization of the scalarized Bi-HYCO functional in the shared-observation setting.

Figure 2. Decentralized realization for fragmented observations. Each local unit updates a copy of the parameter pair using its available observations and the interaction term. The coordinator then averages the updated parameter pairs. When

\lambda_{\rm int}=0

, the explicit state-interaction term is absent, whereas coordinator aggregation remains active.

Metric	$\eta$
Metric	0.00	0.05	0.10	0.15	0.20
$\hat{a}_{1}$	1.971312	1.998491	2.010475	2.001795	1.987880
$\hat{a}_{2}$	1.001103	1.000469	1.001298	0.997622	0.999996
$\delta a_{1}$	0.014344	0.000755	0.005237	0.000897	0.006060
$\delta a_{2}$	0.001103	0.000469	0.001298	0.002378	0.000004
$e_{\gamma}$	0.014386	0.000888	0.005396	0.002541	0.006060

Table 1. Estimated diffusion coefficients and relative errors under increasing observation noise.

Metric	$\eta$
Metric	0.00	0.05	0.10	0.15	0.20
$e_{\rm phy}(\Omega)$	0.0030	0.0030	0.0030	0.0044	0.0032
$e_{\rm syn}(\Omega)$	0.0821	0.0648	0.0614	0.0400	0.0664
$e_{\rm phy}(\Omega_{1})$	0.0013	0.0010	0.0018	0.0022	0.0010
$e_{\rm phy}(\Omega_{2})$	0.0039	0.0040	0.0038	0.0057	0.0042
$e_{\rm syn}(\Omega_{1})$	0.0266	0.0187	0.0150	0.0163	0.0156

Table 2. Relative

L^{2}

reconstruction errors and cross-subdomain error differences across noise levels.

$(\omega_{\mathrm{phy}},\omega_{\mathrm{syn}})$	$\hat{a}_{1}$	$\hat{a}_{2}$	$\delta a_{1}$	$\delta a_{2}$	$e_{\gamma}$
(1.00, 0.00)	1.964579	1.000530	0.017711	0.000530	0.017719
(0.75, 0.25)	2.032572	0.996496	0.016286	0.003504	0.016659
(0.50, 0.50)	1.987880	0.999996	0.006060	0.000004	0.006060
(0.25, 0.75)	2.211544	0.992827	0.105772	0.007173	0.106015
(0.00, 1.00)	1.260271	0.993013	0.369864	0.006987	0.369930

Table 3. Sensitivity of the recovered coefficients to the scalarization weights

(\omega_{\rm phy},\omega_{\rm syn})

when

\eta=0.20

. All remaining hyperparameters are kept fixed.

Figure 3. Solution reconstruction comparison at $\eta=0.00$ . Rows from top to bottom: global solution, zoomed solution (interface region), and absolute error. Columns from left to right: true solution, Bi-HYCO physics, Bi-HYCO synthetic, PINN, and XPINN. The bottom row uses a linear color scale with $v_{\min}=0$ and $v_{\max}$ set to the 98th percentile of all errors.

Figure 4. Gradient magnitude comparison at $\eta=0.00$ . Rows from top to bottom: three-dimensional perspective of the global gradient field, two-dimensional global view, and two-dimensional zoomed view near the interface. Columns from left to right: true solution, Bi-HYCO physics, Bi-HYCO synthetic, PINN, and XPINN. The first row employs a $3D$ surface plot to highlight the spatial distribution of gradient magnitudes.

Metric	Method
Metric	Bi-HYCO	PINN	XPINN
$\hat{a}_{1}$	1.9713	1.0788	1.3579
$\hat{a}_{2}$	1.0011	1.0481	1.0355
$\delta a_{1}$	0.0143	0.4606	0.3210
$\delta a_{2}$	0.0011	0.0481	0.0355
$e_{\gamma}$	0.0144	0.4631	0.3230

Table 4. Comparison of the reconstruction results obtained with Bi-HYCO, PINN and XPINN at noise level

\eta=0.00

$k$	1	2	3	4
$r_{k}$	0.55	0.70	0.58	0.78
$\theta_{k}$	0.20	1.70	3.35	5.00
$A_{k}$	2.6	-2.2	2.4	-2.0

Table 5. Parameters of the four Gaussian vortices composing the initial condition for the annular domain experiments.

Metric	$\eta$
Metric	0.00	0.05	0.10	0.15	0.20
$\hat{\nu}$	1.020e-03	1.191e-03	1.093e-03	1.065e-03	1.155e-03
$\hat{\rho}$	1.00031	1.00986	1.00463	0.99771	0.99864
$\delta\nu$	0.01977	0.19066	0.09254	0.06486	0.15532
$\delta\rho$	0.00031	0.00986	0.00463	0.00229	0.00136
$e_{\gamma}$	0.01978	0.19091	0.09265	0.06490	0.15532

Table 6. Parameter recovery and interaction-point agreement under increasing relative noise

\eta

Metric	Method
Metric	Bi-HYCO	PINN	XPINN
$\hat{\nu}$	0.0010198	0.0009393	0.0009877
$\hat{\rho}$	1.0003081	0.9894614	0.9922372
$\delta\nu$	0.0197740	0.0606880	0.0122906
$\delta\rho$	0.0003081	0.0105386	0.0077628
$e_{\gamma}$	0.0197764	0.0615962	0.0145369

Table 7. Comparison of Bi-HYCO, PINN, and XPINN on the Navier-Stokes inverse problem (noise level

\eta=0.00

Figure 5. Gradient magnitude $|\nabla\omega|=\sqrt{(\partial_{r}\omega)^{2}+(r^{-1}\partial_{\theta}\omega)^{2}}$ at $t=1.5$ . The first row presents a 3D surface view, the second row a 2D top-down view, and the third row a zoomed 2D view of the same central region.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Metric	Full Bi-HYCO	Vanishing interaction, $\lambda_{\rm int}=0$
$\widehat{a}_{1}$	$1.98788$	$1.98213$
$\widehat{a}_{2}$	$0.99999$	$1.00066$
$\delta_{a_{1}}$	$0.00606$	$0.00893$
$\delta_{a_{2}}$	$4.00\times 10^{-6}$	$0.00066$
$e_{\gamma}$	$0.00606$	$0.00896$

Appendix

Table 8. Parameter recovery for the elliptic transmission inverse problem at

\eta=0.20

. The Bi-HYCO values are those reported in Table 1 .

Parameter	Value
Communication rounds $M$	20000
Local updates per round $E_{\text{local}}$	2
Interaction points $H$	1000
Physical observations	800
Synthetic observations	800
Mesh interior points (coarse)	900

Appendix

Table 9. Key hyperparameters for Decentralized Bi-HYCO.

Loss component	PINN	XPINN
Data	10.0	10.0
PDE residual	1.0	1.0
Boundary (Dirichlet)	10.0	10.0
Interface solution jump	5.0	10.0
Interface flux jump	1.0	2.0

Appendix

Table 10. Loss weights for PINN and XPINN (both at

\eta=0.00

Metric	Full Bi-HYCO	Vanishing interaction, $\lambda_{\rm int}=0$
$\widehat{\nu}$	$1.155\times 10^{-3}$	$1.920\times 10^{-3}$
$\widehat{\rho}$	$0.99864$	$1.04749$
$\delta_{\nu}$	$0.15532$	$0.92009$
$\delta_{\rho}$	$0.00136$	$0.04749$
$e_{\gamma}$	$0.15532$	$0.92131$

Appendix

Table 11. Parameter recovery for the annular Navier–Stokes inverse problem at

\eta=0.20

. The Bi-HYCO values are those reported in Table 6 .

Parameter	Value
Communication rounds $M$	150
Local updates per round $E_{\text{local}}$	100
Interaction points $H$	3000
Physical observations	2500
Synthetic observations	2500
Validation points	3000

Appendix

Table 12. Key hyperparameters for the decentralized Bi-HYCO configuration in the annular experiment.

Loss component	PINN	XPINN
Data fitting	10.0	10.0
PDE residual	1.0	1.0
Poisson residual	1.0	1.0
Initial condition	3.0	3.0
Boundary condition (walls)	1.0	1.0
Interface continuity	—	2.0

Appendix

Table 13. Loss weights for PINN and XPINN (annular experiment,

\eta=0.00

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We introduce HyCOP, a modular framework that learns parametric PDE solution operators by composing simple modules (advection, diffusion, learned closures, boundary handling) in a query-conditioned way. Rather than learning a monolithic map, HyCOP learns a policy over short programs - which module to apply and for how long - conditioned on regime features and state statistics. Modules may be numerical sub-solvers or learned components, enabling hybrid surrogates evaluated at arbitrary query times without autoregressive rollout. Across diverse PDE benchmarks, HyCOP produces interpretable programs, delivers order-of-magnitude OOD improvements over monolithic neural operators, and supports modular transfer through dictionary updates (e.g., boundary swaps, residual enrichment). Our theory characterizes expressivity and gives an error decomposition that separates composition error from module error and doubles as a process-level diagnostic.

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