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Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
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Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
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Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
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Mechanistic models are utilized in individual analysis using single-source data, but imperfections arise due to data collection errors, preventing perfect prediction of observed data. The mathematical equation involves known values (Xi), observed concentrations (Ci), measurement errors (εi), model parameters (ϕj), and the related function (ƒi) for i number of values. Different least-squares metrics quantify differences between predicted and observed values. The ordinary least...
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A parallel-plate capacitor with capacitance C, whose plates have area A and separation distance d, is connected to a resistor R and a battery of voltage V. The current starts to flow at t = 0. What is the displacement current between the capacitor plates at time t? From the properties of the capacitor, what is the corresponding real current?
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    Federated learning combined with scientific machine learning (SciML) trains models on distributed, private data for solving differential equations. New federated models (FedPINNs, FedDeepONets) show competitive accuracy against centralized approaches.

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    Area of Science:

    • Scientific Machine Learning (SciML)
    • Federated Learning (FL)
    • Numerical Analysis

    Background:

    • Complex problems governed by partial differential equations (PDEs) often involve distributed or private data, hindering traditional machine learning approaches.
    • Federated learning (FL) offers a decentralized solution for collaborative model training while preserving data privacy, addressing challenges of data silos and transfer limitations.

    Purpose of the Study:

    • To explore the integration of FL and SciML for approximating complex functions and solving differential equations.
    • To propose novel federated models, federated physics-informed neural networks (FedPINNs) and federated deep operator networks (FedDeepONets).
    • To investigate the impact of data heterogeneity on federated model performance and develop theoretical bounds for weight divergence.

    Main Methods:

    • Development of two novel federated models: FedPINNs and FedDeepONets.
    • Implementation of data generation methods to control non-i.i.d. data and utilization of 1-Wasserstein distance to quantify data heterogeneity.
    • Proposal of a weight divergence measure and a theoretical framework for growth bounds in FL.

    Main Results:

    • Federated methods demonstrated superior performance compared to models trained solely on local data.
    • The proposed FedPINNs and FedDeepONets achieved competitive accuracy relative to centralized models trained on all available data.
    • Systematic investigation revealed the relationship between data heterogeneity and federated model performance.

    Conclusions:

    • The integration of FL and SciML provides an effective framework for solving complex problems with distributed and private data.
    • The novel FedPINNs and FedDeepONets models offer a privacy-preserving and efficient alternative to centralized training for function approximation and PDE solving.
    • The study provides a theoretical understanding of weight divergence in FL and its impact on model performance.