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.

翻译：我们提出了一种贝叶斯模型校准框架，用于推断热传导问题中的非线性本构关系，重点研究随温度变化的热导率。该框架整合了基于梯度的优化与不确定性量化（UQ），以解决从瞬态温度测量中估计热导率函数的逆问题。一项关键贡献是提出了一种自适应算法，该算法能序贯优化用于数值模拟的离散化方案及用于表征热导率曲线的模型复杂度。离散化的优化通过最小化损失函数实现，并采用莫罗佐夫偏差原理作为基于不确定性的终止准则。模型复杂度的选择采用平衡数据似然最大化与过度复杂度惩罚的准则。由此，数值偏差与建模偏差的量级始终与测量噪声引入的不确定性保持一致，从而实现了稳健且计算高效的推断。该方法在合成数据与实验数据上均得到验证，表明其能以最少的过拟合和有限的计算成本实现非线性本构模型的精确校准。