Generalized multi-source assimilation: A framework for cross-modal integration and source optimization

This work establishes the theoretical foundations for optimal weight learning in multi-source scientific machine learning. By moving away from empirical, ad-hoc weighting, our framework explores the automated and mathematically sound fusion of diverse numerical methods, neural architectures, and noisy measurements. We analyze a comprehensive spectrum of coupling mechanisms, ranging from unconstrained Softmax parameterization to explicitly constrained formulations like ADMM, as well as single-timescale, two-timescale, and adaptive- ρ Augmented Lagrangian methods. For the Lagrangian frameworks, we derive separate convergence rates for model parameters ( O ( 1 / k ) ) and source weights ( O (1/ k )), proving that time-scale separation ensures robustness against varying source reliability. We further establish specific regularity conditions for ADMM, including rank and strong convexity requirements for variable splitting. For single-timescale Augmented Lagrangian coupling, by contrast, we establish sufficient conditions under which it achieves sublinear convergence governed by an effective condition number that quantifies the curvature imbalance between primal and dual variables. We validate these theoretical insights across a diverse architectural scope, ranging from Fourier Neural Operators (FNO) and DeepONets to Finite Difference schemes, applied to tasks including multi-scale modeling and large-scale data assimilation. Our results provide insight into a fundamental trade-off: Lagrangian and ADMM approaches provide strict constraint satisfaction and stability for physical laws, while Softmax parameterization offers superior computational efficiency for high-dimensional, entropic ensembles.