Novel Deep neural networks for solving Bayesian statistical inverse

Antil, Harbir; Elman, Howard C.; Onwunta, Akwum; Verma, Deepanshu

doi:10.48550/arxiv.2102.03974

Cited by 12 publications

(34 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The DNNs used in this paper have been motivated by the Residual Neural Network (ResNet) architecture. ResNets have been introduced in [9, 24,2] in the context of data/image classification, see also [1] for parameterized PDEs and [7] where the (related) so-called Neural ODE Nets [4] have been used to solve stiff ODEs. The ResNet architecture is known to overcome the vanishing gradient problem, which has been further analyzed using fractional order derivatives in [2].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…• These networks allow for learning both the solution (see (1)) and difference quotients (see (2)). A similar approach to learn the solution in the context of parameterized PDEs has been recently considered in [1]. • Motivated by chemically reacting flows, the goal is to create networks that can learn multiple reactions propagating multiple species.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the proposed DNNs are motivated by Deep Residual Neural Networks (ResNets). See [24,2] for examples of ResNets for classification problems and [1] for an application to parameterized PDEs.…”

Section: Introductionmentioning

confidence: 99%

“…This requires introducing the so-called adjoint equation (also known as back-propagation in machine learning literature). See [24,2,1] for complete details. Having obtained the gradient, an approximate Hessian is computed via the BFGS routine.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Novel DNNs for Stiff ODEs with Applications to Chemically Reacting Flows

Brown¹,

Antil²,

Löhner³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

Chemically reacting flows are common in engineering, such as hypersonic flow, combustion, explosions, manufacturing processes and environmental assessments. For combustion, the number of reactions can be significant (over 100) and due to the very large CPU requirements of chemical reactions (over 99%) a large number of flow and combustion problems are presently beyond the capabilities of even the largest supercomputers. Motivated by this, novel Deep Neural Networks (DNNs) are introduced to approximate stiff ODEs. Two approaches are compared, i.e., either learn the solution or the derivative of the solution to these ODEs. These DNNs are applied to multiple species and reactions common in chemically reacting flows. Experimental results show that it is helpful to account for the physical properties of species while designing DNNs. The proposed approach is shown to generalize well.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Novel DNNs for Stiff ODEs with Applications to Chemically Reacting Flows

Brown¹,

Antil²,

Löhner³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Input layer regularization for magnetic resonance relaxometry biexponential parameter estimation

Rozowski

Palumbo

et al. 2022

Magnetic Reson in Chemistry

View full text Add to dashboard Cite

Many methods have been developed for estimating the parameters of biexponential decay signals, which arise throughout magnetic resonance relaxometry (MRR) and the physical sciences. This is an intrinsically ill‐posed problem so that estimates can depend strongly on noise and underlying parameter values. Regularization has proven to be a remarkably efficient procedure for providing more reliable solutions to ill‐posed problems, while, more recently, neural networks have been used for parameter estimation. We re‐address the problem of parameter estimation in biexponential models by introducing a novel form of neural network regularization which we call input layer regularization (ILR). Here, inputs to the neural network are composed of a biexponential decay signal augmented by signals constructed from parameters obtained from a regularized nonlinear least‐squares estimate of the two decay time constants. We find that ILR results in a reduction in the error of time constant estimates on the order of 15%–50% or more, depending on the metric used and signal‐to‐noise level, with greater improvement seen for the time constant of the more rapidly decaying component. ILR is compatible with existing regularization techniques and should be applicable to a wide range of parameter estimation problems.

show abstract

Neural network representation of time integrators

Löhner

Antil

2023

Numerical Meth Engineering

View full text Add to dashboard Cite

Deep neural network (DNN) architectures are constructed that are the exact equivalent of explicit Runge–Kutta schemes for numerical time integration. The network weights and biases are given, that is, no training is needed. In this way, the only task left for physics‐based integrators is the DNN approximation of the right‐hand side. This allows to clearly delineate the approximation estimates for right‐hand side errors and time integration errors. The architecture required for the integration of a simple mass‐damper‐stiffness case is included as an example.

show abstract

Novel Deep neural networks for solving Bayesian statistical inverse

Cited by 12 publications

References 39 publications

Novel DNNs for Stiff ODEs with Applications to Chemically Reacting Flows

Novel DNNs for Stiff ODEs with Applications to Chemically Reacting Flows

Input layer regularization for magnetic resonance relaxometry biexponential parameter estimation

Neural network representation of time integrators

Contact Info

Product

Resources

About