SELF-VALIDATED PHYSICS -EMBEDDING NETWORK A GENERAL FRAMEWORK FOR INVERSE MODELLING Ruiyuan Kang

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SELF-VALIDATED PHYSICS-EMBEDDING NETWORK: A

GENERAL FRAMEWORK FOR INVERSE MODELLING ∗

Ruiyuan Kang

Department of Mechanical Engineering

Khalifa University

Abu Dhabi, UAE

ruiyuan.kang@ku.ac.ae

Dimitrios C. Kyritsis

Department of Mechanical Engineering, RICH Center

Khalifa University

Abu Dhabi, UAE

dimitrios.kyritsis@ku.ac.ae

Panos Liatsis

Department of Electrical Engineering and Computer Science

Khalifa University

Abu Dhabi, UAE

panos.liatsis@ku.ac.ae

ABSTRACT

Physics-based inverse modeling techniques are typically restricted to particular research ﬁelds,

whereas popular machine-learning-based ones are too data-dependent to guarantee the physical

compatibility of the solution. In this paper, Self-Validated Physics-Embedding Network (SVPEN),

a general neural network framework for inverse modeling is proposed. As its name suggests, the

embedded physical forward model ensures that any solution that successfully passes its validation is

physically reasonable. SVPEN operates in two modes: (a) the inverse function mode offers rapid

state estimation as conventional supervised learning, and (b) the optimization mode offers a way to

iteratively correct estimations that fail the validation process. Furthermore, the optimization mode

provides SVPEN with reconﬁgurability i.e., replacing components like neural networks, physical

models, and error calculations at will to solve a series of distinct inverse problems without pretraining.

More than ten case studies in two highly nonlinear and entirely distinct applications—molecular

absorption spectroscopy and Turbofan cycle analysis—demonstrate the generality, physical reliability,

and reconﬁgurability of SVPEN. More importantly, SVPEN offers a solid foundation to use existing

physical models within the context of AI, so as to striking a balance between data-driven and

physics-driven models.

Keywords Inverse Modelling ·Physics Embedding ·Neural Network

1 Introduction

A forward problem can be deﬁned as ﬁnding the results (observations)

from a given causes (states)

, the solving

processes which formulates the physical models

to map causes (states) to are called forward modelling, i.e.,

F(x)

,which is demonstrated by the blue line in Fig. 1. On the other hand, the counterpart problem that ascertains

the states from the measured observations is named inverse problem, similarly, its solving process is named inverse

modelling. Forward and inverse problems are the opposite sides of scientiﬁc and engineering problems that can be

seen everywhere. For example, two forward problems can be deﬁned as follows: (a), given the concentration and

temperature of the gas cloud of a certain molecule, to calculate its absorption spectroscopy, which is often termed

Molecular Absorption Spectroscopy (MAS) simulation. (b), given the cycle parameters and components parameters of

an aeroengine, to calculate its performance parameters, such as thrust and thrust speciﬁc fuel consumption, which is

often termed aeroengine performance simulation. Their corresponding inverse problems can be deﬁned as: (a), from the

∗Update information:Kang R. et al. updated on Octobor 17,2022 DOI:10.48550/arXiv.2210.06071

arXiv:2210.06071v3 [cs.NE] 17 Jan 2023

Kang R. et al.

molecular absorption spectroscopy measure, to ascertain the concentration and temperature of gas cloud of a speciﬁc

molecule. (b) given the requirement of aeroengine performance, to ascertain the cycle and component parameters which

satisfy the performance requirements, which is usually termed cycle analysis in aeroengine design.

Figure 1: A high-level representation of forward and inverse modelling methods

In general, the methods solving the inverse problem can be categorized into two types: (a) inverse function method, as

the ﬁrst red line shown in Fig. 1. The inverse process is modelled, so that the state corresponding to a given observation

can be directly calculated, i.e., x=G(y)∼

=F−1(y), (b) Optimization method, as the second red line shown in Fig. 1.

The state corresponding to the given observation is ascertained by iterating the estimated state

ˆx

till a similar observation

is acquired from the forward model. i.e., iterate ˆx, till F(ˆx) = y.

Substantial physics-knowledge-based solutions, inverse-function or optimization ones, have been developed for ap-

plications in speciﬁc science and engineering ﬁelds [

], however, they require substantial background

knowledge and expertise. Moreover, the details of their applications constrain their generalization in other ﬁelds.

Conversely, machine-learning-based solutions have been recently gaining popularity, due to their powerful computation

capability, high ﬂexibility, and reduced requirements for background knowledge. In the scope of inverse function

method, a Machine Learning (ML) model G(y)is trained to approximate the inverse function F−1(y)of the physical

forward model [

], by using the information conveyed by a large amount of experimental or

simulated data. Accordingly, for a given observation

, the ML model is expected to give an efﬁcient estimation of the

corresponding state.

However, such estimations of states may be biased, since the inverse problems are usually rank-deﬁcient ill-posed [

which means many states exist for a speciﬁc observation. Fortunately, the forward problems are always well-posed [

Therefore, in the scope of optimization method, ML models are usually trained to be the surrogate models

F(x)

of the

physical forward models

F(x)

[

], which are then used to provide an accurate estimation of the observation

for the given state. Then optimization methods are utilized to iterate the state estimation

ˆx

till the estimated observation

ˆy

from

F(x)

is similar enough to the actual observation

. The introduction of the surrogate model brings two core

beneﬁts. First, it provides for rapid mapping from the space of states to the space of observations, which in turn support

the implementation of time-consuming computations, such as Computational Fluid Dynamics (CFD) [

]. Second,

the inherent differentiability properties of the network assures the appropriateness of gradient-based optimization

algorithms [20].

The main issue with the use of the aforementioned methods is they are purely data-driven. Since there is no guarantee

that the dataset reﬂects the true distributions of the space of states and observations, the resulting ML models cannot

assure learning a sufﬁciently accurate mapping between these two spaces [

]. Accordingly, ML models may not

provide fairly accurate estimations for noisy data or samples outside the dataset, i.e., weak robustness [

] and

generalization [

]. The consequence is that it is not possible to guarantee the reliability of the ML estimations, thus

limiting the application of purely data-driven methods for the solution of the inverse problem.

An alleviation to this problem is injecting physical knowledge within the modelling framework. Examples of such

approaches include the use of PDE-based regularization [

] to reﬂect the physical conservations the model

should obey, or the results from complex physical models to constrain the training direction of the model [

so that a deeper correlation between the ML and physical models can be built. Nevertheless, still no one can assure

data-driven models act exactly as physical models, especially in corner cases or unseen cases. A thorough solution,

which can also be regarded as injecting physical knowledge, is using the forward model of the physical process directly

Kang R. et al.

instead of a data-driven one, but using a network to guide the direction of state search [

], as often done in meta

learning [

]. This kind of method is still in the scope of optimization method, If the physical model embedded is

believed to be trustful, the ﬁnal state estimation can thus be considered to be reliable. However, the network needs to be

pretrained on the designated physical model, which means that the mismatch between the network and the physical

model during testing could lead to incorrect search direction, getting stuck in local optima, and eventually cannot

converge.

In summary, inverse function methods allow for efﬁcient estimations, while optimization methods, coupled with physical

forward models, provide for effective estimations. However, all these ML-based methods require pre-training, which in

turn translates to requirements for preparation time and resources prior to model deployment, and more important, the

concerns in regard to the generalization performance of the model on the testing dataset.

In this work, we propose a general purpose, neural networks-based framework for the solution of inverse problems,

termed Self-Validated Physics-Embedding Network (SVPEN). The principle of SVPEN is to couple both the inverse

function and optimization methods, through embedding physical forward models into neural networks. By utilizing

physical forward model to validate the quality of estimated state, the system ensures that ﬁnal state estimations are

physically reasonable, while the two problem solving modes of inverse function and optimization lend SVPEN with

their advantages of efﬁcient and effective state estimations. Moreover, SVPEN can be deployed without the use of

pre-collected dataset and pre-training, while its structure can be adapted to update the underlying physical/ML models.

In order to demonstrate the advantages of SVPEN, this contribution considers two diverse inverse problems as we

introduced above, namely, retrieval of temperature and gas concentration from MAC, and aeroengine cycle analysis from

performance requirements (code repo:

https://github.com/RalphKang/SVPEN_1.0

). The results demonstrate the

high efﬁciency, effectiveness, ﬂexibility and adaptivity of the proposed framework.

2 SVPEN

The skeleton of Self-Validated Physics-Embedding Network (SVPEN) is demonstrated in Fig. 2. In general, SVPEN is

constituted of inverse function and optimization modes. Once an observation is fed to SVPEN, the inverse function

mode is ﬁrst used to give a quick estimation of the state; meanwhile, an estimation error which reﬂects the quality of

state estimation is also provided. The estimated state is accepted when the estimation error is smaller than an error

threshold

, otherwise, the system automatically switches to the optimization mode. In the optimization mode, the

estimation error is gradually reduced to be smaller than the error threshold

by performing gradient descent on the

estimated state. The control variable, i.e., estimation error, is determined by the physical forward model embedded

in both modes, thereby, assuring the acceptable estimated state is physically reasonable. In this section, we introduce

in detail the design of SVPEN, while in section 2.1, we present the design of inverse function mode. In section 2.2,

attention shifts to the introduction of the optimization mode, and ﬁnally in section 2.3, we explain the cooperation of

these two modes within the structure of SVPEN, and the associated beneﬁts brought about.

2.1 Inverse function mode

The structure of the inverse function mode is schematically shown in Fig. 3, which is mainly constituted of three

computational components, i.e., state estimator

, physical forward model

, and error calculation component

The state estimator is a ML model, which takes the observation

as the input, and transforms it to an estimated state

ˆx

Then the acquired

ˆx

is fed to

to generate an estimated observation

ˆy

. The difference between estimated and given

observations, and even the prior requirement of state can be used to calculate an error

, which assess the quality of

ˆx

The whole process can be expressed into Eqs. 1- 3.

ˆx=G1(y)(1)

ˆy=F(ˆx)(2)

e=E(y, ˆy, ˆx)(3)

According to the functionalities of physical forward model and error calculation component, they can be wrapped

together into an integrated module termed physical evaluation module, which is used to assess the state estimation

quality from a physical perspective.

2.1.1 State estimator

One can recognize that the functionality of the state estimator is not different to those of ML models in the inverse

function method [

], i.e., it is expected to transform observations into accurate estimations of states. In

Kang R. et al.

Figure 2: The skeleton of SVPEN

Figure 3: The schematic of the inverse function mode

order to realize such a purpose, the state estimator must be trained to learn how to map observations to states before its

deployment. Similarly to the inverse function method, the training is also done in a supervised way by utilizing the

observation-state pairs collected from physical experiments or simulation studies. This supervised training process is

termed as pretraining herein, which in order to differentiate it from the online optimization process in optimization

mode (section 2.2).

It is noteworthy that the state estimator must be a differentiable ML model. Indeed, this requirement is the foundation

of the optimization mode, which will be illustrated in section 2.2. Accordingly, a neural-network-type model is used

as default in SVPEN due to its natural differentiability, ﬂexible structure, and extensive research foundation. For

demonstration purposes, two architectures, i.e., VGG-13, a classical CNN architecture, and a multi-layer perceptron

(MLP), are respectively used in the two application cases, presented in sections 3.1 and 3.2.

Kang R. et al.

2.1.2 Physical evaluation module

As previously mentioned, the physical evaluation module consists of two operations. This module takes the given

observation and the estimated state, and then feedbacks with a physics-determined error

to assess the state estimation

quality. The crucial aspect of

is that it reﬂects the discrepancy between the given observation and the estimated obser-

vation corresponding to the estimated state. In general, only physical models of the forward process are recommended

to be used to map the estimated state to its corresponding observation, since the mapping provided by such models is

regarded as physically trustful. The discrepancy

between the given and estimated observations can be calculated

by a variety of distance functions, such as the Manhattan distance, the Euclidean distance, or a monotonic nonlinear

transformation of distance functions, an example of which is used in section 3.1.1.

However, there are instances where only using the

is insufﬁcient, since the inverse problem is often rank-deﬁcient

and ill-posed [

], i.e., multiple estimates of state may lead to similar discrepancies that smaller than the error threshold

. To tackle this issue, prior-knowledge regularization about the observations and states (shown as the dashed line in

Fig. 3) can be utilized as a part of

to further constrain the state estimation. For example, in practice, the feasible

domain of state is usually ascertained. As for the MAS respectively acquired from a ﬂame and ordinary atmosphere, we

do know the temperature retrieved from these two MAC, will fall into the range of a few thousand K and a few hundred

K, respectively. Consequently, a regularization term can be set as shown in Eq. 4:

ereg =M ax(ˆx−xmax,0) + M ax(xmin −ˆx, 0) (4)

Where,

ereg

is the regularization term,

xmax

and

xmin

are the upper and lower boundaries of the feasible domain of

state, respectively. Therefore, the state estimation

ˆx

exceeds the feasible domain will cause extra error. Besides, the use

of traditional L

norms to constrain the magnitude of the state, or the use of PDE-based regularizations to reﬂect the

physical laws [

], etc., can also be applied, depending on the inverse problem to be solved. For instance, the total

error e can be deﬁned as Eq. 5:

e=c0ey+

i=1

ciereg,i (5)

where the

’s are the weights of the discrepancy /regularization terms. The choice of weights affects the evaluation of

the estimated states. For example, for a problem where no regularizations priors are needed, all weight values except

are set to zero. The calculation of

and all individual discrepancy/regularization error terms is performed by the error

calculation component E.

2.2 Optimization mode

However, as we discussed above, the state estimator

cannot be assured to always provide reasonable estimations, as

it is a data-driven model. In cases where the calculated error

is greater than the default error threshold

, the estimated

state is supposed to be far from its true state, and is thus, deemed as unacceptable. Under such conditions, SVPEN

switched to the optimization mode. The purpose of the optimization mode is to utilize the backpropagation of error

to optimize the estimation of state, until error

satisﬁes the error threshold

or the number of iterations reaches the

selected maximum of t.

However, the fact is that not all physical forward models are fully differentiable, which means that the back propagation

route from the error

to the state estimator

is interrupted, as the cross symbol shown in Fig. 4. Using the adjoint state

method [

] or adding perturbations to every entry of the state could support calculation of the vector of derivatives,

however, these require complex transformations or frequent evaluations of the physical model in every iteration, leading

to increased computational requirements.

Figure 4: The dilemma that back-propagation unavailability for not differentiable physical model

In this research, we exploit the differentiability of the network structure of

, by bridging the error

and the state

estimator

through adding another network-type learner

to the original architecture, as shown in Fig. 5. The

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SELF-VALIDATEDPHYSICS-EMBEDDINGNETWORK:AGENERALFRAMEWORKFORINVERSEMODELLINGRuiyuanKangDepartmentofMechanicalEngineeringKhalifaUniversityAbuDhabi,UAEruiyuan.kang@ku.ac.aeDimitriosC.KyritsisDepartmentofMechanicalEngineering,RICHCenterKhalifaUniversityAbuDhabi,UAEdimitrios.kyritsis@ku.ac.aePanosLiatsi...

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