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

Umberto Biccari, Jun Chen, Roberto Morales, Enrique Zuazua

arXiv:2609.06511v1cs.LGmath.OC

TL;DR

Fragmented observations can leave complementary physical and synthetic models with different partial views of a PDE-governed system. Bi-HYCO retains their local objectives and couples predicted states, with scalarized algorithms for shared or fragmented observations. Experiments report improved parameter recovery from state interaction beyond aggregation alone, while the convergence result applies to the shared-observation core.

  • Problem

    The paper asks how complementary models can cooperate when they receive different, possibly disjoint observations and each has only a partial view of the same phenomenon.

    Source: §1 · pp. 1–2

  • Method

    Bi-HYCO preserves separate physical and synthetic observational objectives and couples the models through agreement of predicted states, using weighted scalarization for practical algorithms.

    Source: §1 · p. 2

  • Results

    In fragmented-observation experiments, state interaction can improve parameter recovery beyond parameter aggregation alone; the shared-observation alternating core converges to a mixed critical point under stated assumptions.

    Source: §1 · p. 2, §6 · p. 17

  • Takeaways & Limitations

    Agreement between predicted states can let heterogeneous models use information learned from different observations while preserving their distinct representations.

    Source: §6 · p. 18

    The convergence result covers the deterministic shared-observation core with fixed interaction points, not the decentralized fragmented-observation scheme, whose behavior is investigated numerically.

    Source: §2.5 · p. 5

Abstract

from arXiv · show

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.

1. Introduction

Physical and synthetic representations can offer complementary but imperfect accounts of a system, while fragmented observations leave each with only a partial view. Bi-HYCO preserves their separate observational objectives and couples predicted states, with shared- and fragmented-observation implementations and analyses of convergence and PDE experiments.

1. Introduction

  • Fragmented, heterogeneous observations raise the question of how complementary models can cooperate when each sees only part of a phenomenon.Neither representation is assumed to be the truth, and inverse mappings may lack injectivity or stability.Source: §1 · pp. 1–2
  • Unlike physics-informed approaches or PDE-constrained estimation that combine constraints and data in a representation or optimize a numerical forward model, HYCO preserves distinct physical and synthetic representations that exchange predicted states.Original HYCO also introduced state-level interaction and conceptually permits heterogeneous observations.Source: §1 · pp. 1–2
  • Bi-HYCO retains each model’s observational objective and uses state agreement to make their two-criterion trade-off explicit, with weighted scalarization for practical algorithms.This preserves state-level interaction while making the model-specific objectives explicit.Source: §1 · p. 2
  • The shared-observation regime uses a weighted scalarization and alternating scheme, for which the whole sequence converges to a mixed critical point under stated assumptions and fixed interaction points.The fragmented-observation regime instead uses distinct datasets and decentralized parameter updates coordinated by aggregation while retaining state interaction.Source: §1 · p. 2
  • Experiments study elliptic transmission and Navier-Stokes parameter identification, assessing reconstruction, noise and scalarization effects, state-interaction ablations, PINN/XPINN references, and computational cost.The problems contrast stationary and evolutionary dynamics, linear and nonlinear models, and interface and annular geometries.Source: §1 · p. 2
  • Bi-HYCO lets distinct physical and synthetic models transfer information through agreement of predicted states without centralizing their observation sets.The interaction points are unlabeled and provide no measurements.Source: §1 · p. 2

2. Bi-HYCO for PDE Parameter Identification with Fragmented Observations

Bi-HYCO retains physical and synthetic PDE representations with separate observational objectives, coupling their predicted states at unlabeled points. It provides shared-observation alternating minimization and a decentralized realization for fragmented observations.

2.1. Physical and synthetic representations and fragmented observations.

  • Physical states come from a numerical PDE solver, while the synthetic representation is parameterized and realized as a finite-dimensional neural ansatz in the numerical examples.The observation operators may coincide, overlap, or probe disjoint regions or time intervals.Source: §2.1 · p. 3

2.2. Unlabeled state interaction.

  • At common interaction locations, the two predicted states are compared without adding measured values or physical observations.Interaction proceeds in communication rounds, each comparing current states at shared locations.Source: §2.2 · p. 3

2.3. Bi-objective formulation, scalarization, and Pareto interpretation.

  • The formulation keeps the physical and synthetic observation-specific criteria explicit, and positive-weight global minimizers of their weighted scalarization are Pareto optimal.The convergence result establishes mixed criticality, not global minimization; nonconvex weighted sums may miss unsupported Pareto points.Source: §2.3 · p. 4

2.4. Shared-observation alternating core.

  • With shared observations, Bi-HYCO alternates updates of the physical and synthetic parameter blocks; the stated convergence result is limited to fixed interaction points in this setting.When λint = 0 and both block weights are positive, the two blocks decouple.Source: §2.4 · p. 4, §2.5 · p. 5

2.5. Fragmented-observation decentralized realization.

  • With fragmented observations, separate local units update parameter copies using their available data and state interaction, then a coordinator aggregates the copies.Setting λint = 0 removes state coupling while preserving local optimization and coordinator aggregation.Source: §2.5 · pp. 4–5, Figure 2 · p. 6

2.6. Aggregate interpretation.

  • The decentralized method has FedAvg-like parameter aggregation, but its local objectives also share state-level coupling between the representations.The aggregate functional combines both data-dependent losses with the state-interaction term.Source: §2.6 · pp. 5–6, §2.5 · p. 5
  • With one local update, aggregation equals a gradient step for the aggregate functional; with multiple updates, local iterates follow different trajectories before aggregation.The FedAvg-style interpretation is heuristic, and a rigorous convergence analysis adapted to state interaction is omitted.Source: §2.6 · pp. 6–7

3. Convergence analysis of the alternating scheme

For the deterministic shared-observation alternating scheme with fixed interaction points, the paper proves whole-sequence convergence to a mixed critical point under assumptions (A1)–(A5). The result is an optimization-convergence guarantee for this core, not a guarantee of global or Pareto optimality or convergence of the decentralized fragmented-observation realization.

3. Convergence analysis of the alternating scheme

  • The limit is mixed critical: physical stationarity includes the normal cone to the constraint set, while synthetic stationarity uses a partial Clarke subdifferential.This definition accommodates nonsmooth synthetic losses.Source: §3 · p. 7
  • The guarantee relies on regularity and Kurdyka-Lojasiewicz assumptions, a bounded initial sublevel set, and step-size and descent conditions in (A1)–(A5).The theorem assumes a closed convex physical constraint set and locally Lipschitz synthetic loss.Source: §3 · p. 8
  • Under assumptions (A1)–(A5), with shared observations and fixed interaction points, the entire alternating sequence converges to a mixed critical point.The analysis concerns the scalarized shared-observation iteration.Source: §3 · pp. 7–9
  • Sufficient decrease and finite length of the iterates underpin the proof of whole-sequence convergence.Finite total step length makes the sequence Cauchy; its limit is mixed critical.Source: §3 · p. 9
  • When the interaction weight is zero, the alternating method reduces to two independent problems.Source: §3 · p. 9
  • The theorem covers only the deterministic shared-observation core with fixed interaction points; Adam or L-BFGS use alone does not ensure its descent and relative-error conditions.It neither establishes global or Pareto optimality nor covers the decentralized fragmented-observation realization.Source: §3 · p. 10

4. Elliptic transmission inverse problem: interaction and Bi-Objective trade-offs

This section tests Bi-HYCO on an elliptic transmission inverse problem with spatially fragmented observations, examining parameter and state recovery, noise and scalarization effects, and comparison with PINN/XPINN.

4.1. PDE model and inverse problem.

  • The elliptic experiment targets recovery of two piecewise-constant diffusion coefficients from spatially fragmented observations across an irregular domain with an internal interface.The two local data-holding units receive measurements in separate subdomains.Source: §4 · p. 10, §4.1 · p. 10

4.2. Bi-HYCO implementation.

  • The implementation specifies communication rounds, local updates, learning-rate schedules, architecture, and interaction-point sampling; details are reported in Appendix A.2.2.The experiment assesses recovery from fragmented observations and compares reconstruction with PINN/XPINN references.Source: §4 · p. 10, §4.2 · p. 11

4.3. Sensitivity to observation noise.

  • Across five fixed noise realizations, joint parameter error stays below 1.5%, with a minimum of 0.089% at η = 0.05; the nonmonotone pattern is descriptive, not evidence that noise improves recovery.The relative error in a2 stays below 2.4 × 10^-3, while the error in a1 ranges from 0.08% to 1.43%.Source: §4.3 · p. 11
  • The solver-based reconstruction has global relative error of order 10^-3, versus order 10^-2 for the synthetic state; the physical reconstruction has comparable accuracy across subdomains.The synthetic error difference is negative in the reported runs, indicating smaller error in Ω1 than in Ω2; these are spatial diagnostics, not out-of-sample estimates.Source: §4.3 · pp. 11–12
  • At η = 0.20, recovery is sensitive to scalarization: small parameter errors occur when the physical criterion has sufficient weight, while results deteriorate as the synthetic criterion dominates.The remaining algorithmic parameters are held fixed in the weight-sensitivity experiment.Source: §4.3 · p. 12, Table 3 · p. 12
  • With interaction removed but data distribution, local optimization, and coordinator aggregation unchanged, joint parameter error rises from 0.00606 to 0.00896, about 1.48-fold.This ablation is at η = 0.20.Source: §4.3 · p. 12

4.4. Comparison with Physics-Informed Neural Network Variants.

  • In the noise-free realization, Bi-HYCO has the lowest parameter-recovery error and its solver-based state the lowest global state error, at 2.12–2.45 times PINN/XPINN wall-clock time.Because the physical state is generated by a finite-element solver after coefficient identification, the comparison does not isolate optimization from solver-based state reconstruction.Source: §4.4 · pp. 12–13, Figure 3 · p. 13

5. Navier-Stokes Parameter Identification: A Nonlinear Fragmented-Observation Test

The annular Navier-Stokes test examines parameter and state reconstruction from spatially fragmented vorticity observations. Bi-HYCO, PINN, and XPINN show different strengths across parameter and state metrics, while a matched ablation tests explicit state interaction.

5.1. PDE model, observations and Bi-HYCO setup.

  • The annular inverse problem recovers viscosity and forcing amplitude from vorticity observations split between alternating angular sectors.The reference domain has inner and outer radii 0.25 and 1.0.Source: §5 · p. 13, §5.1 · p. 15

5.2. Sensitivity to observation noise.

  • At η = 0.20, eγ rises from 0.15532 to 0.92131 when explicit interaction is removed, with fragmentation, local optimization, and aggregation unchanged.The matched ablation isolates the interaction term's contribution in this run, not general robustness or identifiability.Source: §5.2 · p. 16

5.3. Comparison with Physics-Informed Neural Network variants.

  • The solver-based Bi-HYCO state is most accurate, and the synthetic component retains nontrivial accuracy in the sector without direct observations.The comparison alone does not separate state interaction from parameter aggregation.Source: §5.3 · p. 16
  • Solver-based Bi-HYCO errors are generally smaller than PINN and XPINN errors, while XPINN shows oscillations near some angular-sector interfaces.Figure 5's gradient peaks for solver-based Bi-HYCO are closer in location and amplitude to the reference, as a qualitative diagnostic.Source: §5.3 · p. 16, Figure 5 · p. 17
  • XPINN has the smallest joint parameter error, while Bi-HYCO has the most accurate forcing-amplitude estimate and smallest global state error.The comparison is for the reported noise-free realization.Source: §5.3 · p. 16, Table 7 · p. 17

6. Conclusions, discussion and perspectives

Bi-HYCO combines physical and synthetic representations with separate observation objectives and unlabeled state-matching points. The paper reports convergence for the shared-observation alternating scheme and numerical evidence that state interaction can improve parameter recovery beyond aggregation alone.

6. Conclusions, discussion and perspectives

  • Bi-HYCO retains each model's observation objective and couples predicted states at unlabeled points; weighted scalarization provides a practical balance between criteria.For the alternating scheme with shared observations, the whole sequence converges to a mixed critical point.Source: §6 · p. 17
  • Predicted-state agreement lets heterogeneous models use information learned from different observations while preserving their distinct representations.The framework is intended for settings where different observations are available to different models.Source: §6 · p. 18
  • Future work includes decentralized convergence analysis, parameter-identifiability effects of state interaction, stability estimates involving Lint, and adaptive interaction-point selection.The paper also identifies higher-dimensional parameter spaces and multiple models or interfaces as computational extensions.Source: §6 · p. 18

Appendix A. Experimental implementation details

Appendix A records experimental implementation and reproducibility material. It distinguishes these supporting details from the mathematical validity of the main convergence theorem.

Appendix A. Experimental implementation details

  • Appendix A provides implementation, reproducibility, ablation, and diagnostic material for the experiments.These materials support reproduction and interpretation but are not required for the main convergence theorem's mathematical validity.Source: Appendix A · p. 18
  • The complete proof of Theorem 3.4 is in the refereed main manuscript, not this appendix.Appendix A.1-A.3 instead records computing environment, experiment settings, ablations, and implementation details.Source: Appendix A · p. 18
  • The simulations ran on Windows 11 with an Intel Core Ultra 5 225H processor and 32 GB of RAM.The implementation used Python 3.13.5 in the Anaconda distribution.Source: §A.1 · p. 18

A.2.1. Vanishing-interaction experiment.

The vanishing-interaction experiment keeps the fragmented-data procedure unchanged while removing explicit state coupling. In the reported noisy elliptic realization, parameter errors increase, especially for coefficient a1.

A.2.1. Vanishing-interaction experiment.

  • At η = 0.20, the ablation removes Lint while keeping the data distribution, local optimization, and coordinator aggregation unchanged.It serves as an internal reference for the explicit interaction term's role.Source: §A.2.1 · p. 18
  • At η = 0.20, eγ increases from 0.006060 to 0.00896, while the relative errors in a1 and a2 rise from 0.006060 to 0.00893 and 4 × 10^-6 to 6.6 × 10^-4.The joint error increases by approximately a factor of 1.48.Source: §A.2.1 · p. 19
  • Removing interaction deteriorates both coefficient estimates in this realization, more markedly for a1, but does not establish general robustness or identifiability.By joint error, Bi-HYCO gives the more accurate parameter reconstruction at η = 0.20.Source: §A.2.1 · p. 19

A.2.3. PINN and XPINN implementation details.

The implementation combines data fitting, PDE and boundary constraints, and interface continuity, with weights chosen to balance the loss terms. The experiments use fixed-seed computations and the same meshes and other settings as Bi-HYCO.

A.2.3. PINN and XPINN implementation details.

  • The loss includes data fitting, PDE residual, boundary conditions, and interface continuity for solution and flux, with weights chosen to balance the terms.The coefficients use the same softplus parameterization as Bi-HYCO.Source: §A.2.3 · p. 20
  • Table 10 summarizes the loss weights.Its caption specifies PINN and XPINN at η = 0.00.Source: Table 10 · p. 20, §A.2.3 · p. 20
  • Computations use fixed random seed 7, while mesh, observations, noise generation, and evaluation metrics match the Bi-HYCO settings.The optimization and evaluation meshes are those described in Section 4.Source: §A.3 · p. 20, §A.2.3 · p. 20

A.3. Navier-Stokes experiment.

The Navier–Stokes experiment tests whether state interaction adds to coordinator aggregation in a noisy fragmented-observation setting. The aggregation-only comparison shows worse parameter recovery, but it is evidence from one noise realization and parameter configuration.

A.3.1. Vanishing-interaction experiment.

  • The ablation sets λint=0 at η=0.20 while retaining the data distribution, local optimization, and coordinator aggregation.It compares this aggregation-only configuration with full Bi-HYCO and ground-truth parameter values.Source: §A.3.1 · p. 20
  • Table 11 reports substantially worse parameter recovery without interaction, especially for viscosity; forcing-amplitude and joint parameter errors also increase.The table reports the annular Navier–Stokes inverse problem at η = 0.20.Source: Table 11 · p. 20, §A.3.1 · p. 20
  • This matched comparison is numerical evidence from one noise realization and parameter configuration, not a general robustness statement.The authors attribute a significant contribution to parameter recovery to explicit agreement between physical and synthetic states.Source: §A.3.1 · p. 20
  • Bi-HYCO uses a Fourier-feature synthetic network and a coarser polar-grid physical solver; interaction points are resampled each communication round.The synthetic network has four width-80 tanh layers; the physical grid has 64 radial and 128 angular cells, and the vanishing-interaction run changes λint from 2 to 0.Source: §A.3.2 · p. 20, Table 12 · p. 21
  • Bi-HYCO, PINN, and XPINN share feasible parameter bounds and an effective initialization of (ν0, ρ0) = (5 × 10−3, 0.5).The common effective starting point follows projection onto the feasible box.Source: §A.3.3 · p. 21
  • PINN and XPINN use identical data and evaluation conditions; XPINN divides the angular domain among four independent networks.Both use 10,000 Adam epochs, and XPINN includes interface-continuity losses for ψ and ω.Source: §A.3.4 · pp. 21–22, Table 13 · p. 21

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