7.3NAJul 14
Stochastic Finite Volume Approximation with Clustering in the Parameter Space for the Forward Uncertainty Quantification of Differential Equations with Random ParametersZhao Zhang, Mengyao Xia, Na Ou
The uncertainty quantification (UQ) for mathematical models with random parameters is important for many science and engineering problems. Forward UQ quantifies the impact of random parameters on the output of system. In the current study, we propose a new stochastic finite volume (SFV) scheme by combining SFV with clustering algorithm in the parameter space such that each cluster can be regarded as a finite volume with implicit boundaries. The advantage of SFV is that no specific form of the random variable is required such that discontinuous solutions and sharp interfaces can be accurately approximated. Compared to classic SFV based on structured grids, the new SFV-cluster scheme extends SFV to parameter spaces of higher dimensions. For demonstration and validation, we present the construction of SFV schemes for the Kraichnan-Orszag three-mode problem and the Buckley-Leverett equation. The error analysis of SFV is extended and the new algorithm is validated by numerical experiments.
1.2NADec 2, 2025
A Discrete Neural Operator with Adaptive Sampling for Surrogate Modeling of Parametric Transient Darcy Flows in Porous MediaZhenglong Chen, Zhao Zhang, Xia Yan et al.
This study proposes a new discrete neural operator for surrogate modeling of transient Darcy flow fields in heterogeneous porous media with random parameters. The new method integrates temporal encoding, operator learning and UNet to approximate the mapping between vector spaces of random parameter and spatiotemporal flow fields. The new discrete neural operator can achieve higher prediction accuracy than the SOTA attention-residual-UNet structure. Derived from the finite volume method, the transmissibility matrices rather than permeability is adopted as the inputs of surrogates to enhance the prediction accuracy further. To increase sampling efficiency, a generative latent space adaptive sampling method is developed employing the Gaussian mixture model for density estimation of generalization error. Validation is conducted on test cases of 2D/3D single- and two-phase Darcy flow field prediction. Results reveal consistent enhancement in prediction accuracy given limited training set.