Hai Van Nguyen
Methodology
- This work aims to solve shock-type problems with machine learning.
Figure 1: The schematic of DGNet network architecture.
Numerical results
- We work on 2D Euler equations
\[\begin{align*}
\frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} &= 0 \\
\frac{\partial (\rho u)}{\partial t} + \frac{\partial (\rho u^2 + p)}{\partial x} + \frac{\partial (\rho u v)}{\partial y} &= 0 \\
\frac{\partial (\rho v)}{\partial t} + \frac{\partial (\rho u v)}{\partial x} + \frac{\partial (\rho v^2 + p)}{\partial y} &= 0 \\
\frac{\partial E}{\partial t} + \frac{\partial (u(E + p))}{\partial x} + \frac{\partial (v(E + p))}{\partial y} &= 0
\end{align*}\]
- where \(E\) is the total energy per unit volume:
\(E = \frac{p}{\gamma - 1} + \frac{\rho}{2}(u^2 + v^2)\)
Problem 1. Airfoil NACA0012
- Training data is generated from Airfoil AoA = 3 and Mach = 0.8, in time interval [0,1.2]s
- Test data is generated from Airfoil AoA = 3 and Mach = 0.8 and Airfoil AoA = 5 and Mach = 1.2 for time interval [0, 7.5]s
Figure 2: (Airfoil) Airfoil configuration AoA-3.
Figure 3: (Airfoil) predictions by DGNet for Airfoil NACA0012 of AoA = 5 and Mach = 1.2.
Problem 2. Euler configurations 6 & 12
- Training data is generated from Euler configuration 6 with time interval [0,0.16]s
- Test data is generated from Euler configuration 6 for time interval [0, 0.8]s and Euler configuration 12 for time interval [0, 0.25]s
Figure 4: (Euler configurations) Information settings.
Figure 5: (Euler configurations) predictions by DGNet for configuration 6.
Figure 6: (Euler configurations) predictions by DGNet for configuration 12.
Problem 3: Double Mach Reflection
- Training data is generated with time interval [0,0.02]s
- Test data is generated with time interval [0, 0.25]s
Figure 7: (Double Mach Reflection) model.
Figure 8: (Double Mach Reflection) predictions by DGNet.
Problem 4. Forward facing corner
- Training data is generated from Model 1 with time interval [0,1]s
- Test data is generated from Model 1 and Model 2 for time interval [0,4]s
Figure 9: (Forward-facing corner) Model 1 and Model 2.
Figure 10: (Forward-facing corner) predictions by DGNet for Model 1.
Figure 11: (Forward-facing corner) predictions by DGNet for Model 2.
Major Activities
Last year, we have been working on the the theorectical part for pure machine learning (nDNN) and model-constrained deep learning approaches (mcDNN). The numerical results are provided for linear 1D inverse deconvolution problem, and 2D Heat equation. The mcDNN requires less training data to achieve the same accuracy level compared to nDNN. It is due to the fact that, for linear case, mcDNN has the data-driven form of Tikhonov inverse solver framwork, for non-linear problem, mcDNN is reinforced by the model-constrained term, and thus encodeing the physic. This year we develop the Tikonov neural network approach (TNet) and apply all approaches to more complicated problems (2D Burgers’ equations and 2D Navier-Stokes equation). We present and provide intuitions and rigorous results for TNet. The advantgeous features of Tnet are (1) requires a few observation samples and no need for true parameter of interest (PoI); (2) recovering exactly Tikhonov framework in the case of linear inverse problem (3) faster convergence rate (with fewer training samples) to Tikhonov’s accuracy level than mcDNN and nDNN.
More detail about this work can be found at https://arxiv.org/abs/2105.12033.
Figure 1: Model-constrained neural network architecture mcDNN. The observables y is fed into the neural network Ψ. The parameter u∗ predicted by the netork is pushed through the PtO map G to generate the corresponding predicted observations y∗. Both predicted parameters and observations are compared with ground truth u and y, respectively, to provide the mean-square error in the loss function L..
Significant Results
Figure 2: 1D Linear deconvolution problem The average relative error at the optimal regularization parameter versus the training size for nDNN, mcDNN, TNet, and Tikhonov methods. The result for Case II (data augmentation technique) is on the left figure for training size up to $5000$, and a similar result is shown on the right figure for Case III (all distinct samples data).
Major Activities
This work has been completed during this year and is under rereview in International Journal of Computational Fluid Dynamics [paper]. The abtract: Real-time accurate solutions of large-scale complex dynamical systems are in critical need for control, optimization, uncertainty quantification, and decision-making in practical engineering and science applications. This paper contributes in this direction a model-constrained tangent slope learning (mcTangent) approach. At the heart of mcTangent is the synergy of several desirable strategies: i) a tangent slope learning to take advantage of the neural network speed and the time-accurate nature of the method of lines; ii) a model-constrained approach to encode the neural network tangent with the underlying governing equations; iii) sequential learning strategies to promote long-time stability and accuracy; and iv) data randomization approach to implicitly enforce the smoothness of the neural network tangent and its likeliness to the truth tangent up second order derivatives in order to further enhance the stability and accuracy of mcTangent solutions. Both semiheuristic and rigorous arguments are provided to analyze and justify the proposed approach. Several numerical results for transport equation, viscous Burger’s equation, and Navier-Stokes equation are presented to study and demonstrate the capability of the proposed mcTangent learning approach.
Figure 1: The schematic of mcTangent network architecture for S = 1.
Significant Results
Major Activities
Last year, we have been working the theorectical part and most of promising numerical results are produced. This year we provided complete proof for the theorectical part and wrapped up the work for publication. This work has been published [paper]. The paper presents a regularization framework that aims to improve the fidelity of Tikhonov inverse solutions. At the heart of the framework is the data-informed regularization idea that only data-uninformed parameters need to be regularized, while the data-informed parameters, on which data and forward model are integrated, should remain untouched. We propose to employ the active subspace method to determine the data-informativeness of a parameter. The resulting framework is thus called a data-informed (DI) active subspace (DIAS) regularization. Four proposed DIAS variants are rigorously analyzed, shown to be robust with the regularization parameter and capable of avoiding polluting solution features informed by the data. They are thus well suited for problems with small or reasonably small noise corruptions in the data. Furthermore, the DIAS approaches can effectively reuse any Tikhonov regularization codes/libraries. Though they are readily applicable for nonlinear inverse problems, we focus on linear problems in this paper in order to gain insights into the framework. Various numerical results for linear inverse problems are presented to verify theoretical findings and to demonstrate advantages of the DIAS framework over the Tikhonov, truncated SVD, and the TSVD-based DI approaches