We present a Bayesian model calibration framework for inferring nonlinear constitutive relationships in heat conduction problems, with a focus on temperature-dependent thermal conductivity. The proposed framework integrates gradient-based optimization and uncertainty quantification (UQ) to address the inverse problem of estimating the conductivity function from transient temperature measurements. A key contribution is an adaptive algorithm that sequentially refines both the numerical discretization for model simulation, and the model complexity used to represent the conductivity curve. The discretization is optimized through the minimization of a loss function, and Morozov's discrepancy principle is used as an uncertainty-motivated stopping criterion. The model complexity is selected using an approach that balances maximizing the likelihood of the data with penalizing excessive model complexity. As a result, the numerical and modeling biases remain of the same order as the uncertainty imposed by the measurement noise, leading to robust and computationally efficient inference. The methodology is demonstrated on both synthetic and experimental data, showing that it enables accurate calibration of nonlinear constitutive models with minimal overfitting and limited computational cost.

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