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

arXiv: https://arxiv.org/abs/2609.06511 · HTML: https://mlsanity.com/paper/2609.06511/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 30 references that are in the map.
- First posted on arXiv 6 Sep 2026.

## Builds on (most cited first, 7 of 7)

- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [On the Convergence of FedAvg on Non-IID Data](https://mlsanity.com/paper/1907.02189/) (2019)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [Combining physics-based and data-driven models: advancing the frontiers of research with Scientific Machine Learning](https://mlsanity.com/paper/2501.18708/) (2025)
- [HYCO: Hybrid-Cooperative Learning for Data-Driven PDE Modeling](https://mlsanity.com/paper/2509.14123/) (2025)
- [HYCO: A Formalism for Hybrid-Cooperative PDE Modelling](https://mlsanity.com/paper/2602.23859/) (2026)

## Abstract (arXiv)

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.