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Physics-Infused Machine Learning Based Prediction

of VTOL Aerodynamics with Sparse Datasets

Manaswin Oddiraju , Divyang Amin , Michael Piedmonte, Souma Chowdhury

Abstract

Complex optimal design and control processes often require repeated evaluations of expensive objective functions and consist

of large design spaces. Data-driven surrogate models such as neural networks and Gaussian processes provide an attractive

alternative to expensive simulations and are utilized frequently to represent these objective functions in optimization. However,

pure data-driven models, due to a lack of adherence to basic physics laws and constraints, are often poor at generalizing

and extrapolating. This is particularly the case, when training occurs over sparse high-ﬁdelity datasets. A class of Physics-

infused machine learning (PIML) models integrate ML models with low-ﬁdelity partial physics models to improve generalization

performance while retaining computational efﬁciency. This paper presents two potential approaches for Physics infused modelling

of aircraft aerodynamics which incorporate Artiﬁcial Neural Networks with a low-ﬁdelity Vortex Lattice Method model with blown

wing effects (BLOFI) to improve prediction performance while also keeping the computational cost tractable. This paper also

develops an end-to-end auto differentiable open-source framework that enables efﬁcient training of such hybrid models. These two

PIML modelling approaches are then used to predict the aerodynamic coefﬁcients of a 6 rotor eVTOL aircraft given its control

parameters and ﬂight conditions. The models are trained on a sparse high-ﬁdelity dataset generated using a CHARM model. The

trained models are then compared against the vanilla low-ﬁdelity model and a standard pure data-driven ANN. Our results show

that one of the proposed architecture outperforms all the other models and at a nominal increase in execution time. These results

are promising and pave way to PIML frameworks which are able generalize over different aircraft and conﬁgurations thereby

signiﬁcantly reducing costs of design and control.

I. INTRODUCTION

Real world optimal design and control problems consist of optimizations with large design spaces and computationally

expensive objective functions and therefore require efﬁcient and accurate models of complex physical systems. There are

different proposed approaches to render these problems computationally feasible, such as the use of surrogate models, reduced

order modelling methods and multi-ﬁdelity optimization techniques. The use of data-driven surrogate models for optimization

[1] is a popular approach to reduce the computational burden and render the process tractable. However, using pure data

driven models poses additional challenges in the form of poor explainability and usually these models require large amounts of

data either from expensive high-ﬁdelity analysis or experiments in order to achieve sufﬁcient accuracy. For many engineering

design problems however fast running physics-based models are available, but they usually trade-off accuracy for performance.

Therefore, in this paper, we propose two Physics Informed Machine Learning (PIML) models which combine data driven

Artiﬁcial Neural Networks (ANNs) with low-ﬁdelity physics models and augment the performance of the physics models

while only needing a sparse high-ﬁdelity dataset. We also develop an end-to-end auto-differentiable physics framework to

enable efﬁcient training of such hybrid architectures and apply them to model the aerodynamics of a 6 rotor eVTOL aircraft.

The rest of this section brieﬂy covers existing techniques for augmenting low-ﬁdelity physics models using high-ﬁdelity data,

physics informed machine learning architectures before stating our research objectives.

Conventional methods for enhancing low-ﬁdelity computational models with high-ﬁdelity or experimental data encompass

techniques like Bayesian Inference [2], [3], Gradient-Based and Gradient-Free Optimization [4], [5], among others. The primary

goal of these techniques is to determine the optimal parameters of the low-ﬁdelity model in order to improve its overall

performance. However, due to the complexity of the physical phenomena often modeled by these techniques, identifying a

static set of parameters for a dataset might be inadequate and may not fully capitalize on the potential accuracy improvements

offered by a sparse high-ﬁdelity dataset. On the other hand, data-driven models, due to their near-instantaneous run times,

are being utilized frequently for speculating the behavior of complex systems in domains such as mechanical systems [6],

[7], robotics [8]–[10], and energy forecasting [11]. Models such as Neural Networks, Gaussian Processes etc. are showing

competitive prediction accuracy on some problems in this domain. However, they underperform at extrapolating and generalizing

[12], [13], especially when trained with small or sparse data sets [14]. This can be attributed to a lack of adherence to basic

laws of physics. Additionally, they also exhibit challenging-to-interpret black-box behavior and sensitivity to noisy data [15].

In lieu of these problems, signiﬁcant research efforts are being directed towards ”hybrid models” or ”physics-infused” models.

These models generally integrate data-driven and low-ﬁdelity physics models in order to make predictions computationally

feasible and accurate.

There are different classes of physics-infused neural networks; although almost all of them make use of a data-driven model

and a physics model, they differ in the architectures. Hybrid ML architectures can broadly be classiﬁed into serial[16]–[19]

and parallel [20]–[22] architectures. Serial architectures typically have the data-driven models set in sequence with the partial

physics model or used to tune the partial physics model parameters, while parallel architectures usually contain additive or

This work was presented at the AIAA Aviation 2023 Forum.

Copyright @ Souma Chowdhury

1

arXiv:2307.03286v1 [cs.CE] 6 Jul 2023

multiplicative ensembles of partial physics and data-driven ML models [23]. Several such hybrid PIML architectures have

been reported in the literature in the past few years [24]–[36], spanning over a wide range of applications such as in modeling

dynamic systems, cyber-physical systems, robotic systems, ﬂow systems and materials behavior, among others. The OPTMA

model [37], is a physics-infused machine learning model which combines an artiﬁcial neural network with a partial physics

model in order to make predictions. Earlier applications of this framework on acoustics problems have shown that it generalizes

well and can even extrapolate successfully. This architecture primarily works by using the data-driven model -in this case the

transfer network; to map the original inputs to the inputs of the partial physics model so as to make its output match that of the

high-ﬁdelity model. Hence, the training process is just the transfer network learning this mapping. And since the intermediate

parameters are open and interpretable to domain experts (as they are inputs to partial physics models), the OPTMA model is

less of a black box and more understandable when compared to pure data-driven models.

In all of these sequential hybrid-ML models however, the presence of an external partial physics model increases the

complexity and cost of the training process. Previously[38], this hurdle was overcome by programming custom loss functions

which include the partial physics in PyTorch[39] to enable backpropagation. However, this approach may not be feasible in

all domains as PyTorch is not optimal for general purpose scientiﬁc computing (especially numerical methods). Therefore, in

our goal to make OPTMA more user-friendly and computationally tractable, we use Google JAX [40] to auto-differentiate

the partial physics model. JAX is capable of forward and backward mode auto-differentiation and works on codes written in

python and is also easy to integrate with the transfer network written in PyTorch. This is a much more generalizable and

scalable approach to creating computationally efﬁcient PIML frameworks.

During the process of aircraft design, and eventually the aircraft controller design, engineers often encounter the earlier stated

predicament of selecting appropriate methods to model the aircraft. They may need to balance the beneﬁts and drawbacks of

using a method that is highly accurate, albeit costly and time-consuming, against a faster and more economical approach that

might not provide the same level of precision.The complexity of this problem is magniﬁed when engineers are tasked with

designing more challenging-to-control aircraft than traditional ﬁxed-wing varieties. A prime example is the Vertical Takeoff &

Landing (VTOL) aircraft. As the aircraft designs become more complex, the modeling demands can escalate exponentially. In

the case of VTOL, it is essential to model the aircraft dynamics across all phases: from hover to transition, and ﬁnally into

cruise. Such aircraft also exhibit more intricate aerodynamics, including phenomena like blown-wing effects. There is typically

a greater number of degrees of freedom, which further escalates the complexity of designing robust controllers for VTOL

aircraft. Given the numerous beneﬁts and strengths of physics-infused modeling, and the necessity for fast accurate models

for aircraft design and control, it is evident that PIML models are particularly well-suited for modeling the aerodynamics of

an aircraft.

Therefore, in this paper, we propose two physics informed modelling approaches to augment a low-ﬁdelity VLM model

with a sparse high-ﬁdelity dataset. One of the approaches, called PIML-A is an extension of the OPTMA architecture. The

other architecture, named PIML-B is a comparatively simpler ensemble model that learns the error in the low-ﬁdelity physics

models. The architectures are aimed at improving the accuracy of the low-ﬁdelity physics models while only using a sparse

high-ﬁdelity dataset to keep training costs low. Our research objectives are as follows:

1) Create a Vortex Lattice Method (VLM) model with auto-differentiation capabilities.

2) Develop different PIML architectures seeking to provide better accuracy / cost trade-offs than pure Low-Fidelity and

pure data-driven models.

3) Apply the above architectures to model the control inputs to force and moment coefﬁcients of a eVTOL aircraft.

4) Compare the performance and computational cost of the PIML models with the baseline low-ﬁdelity and data-driven

models.

The remainder of this paper is laid out as follows: SectionII contains a description of the two PIML models framework and

their components in detail. SectionIII explains our case study and the models that we are using before moving on to SectionIV

which contains our results and discussion. Finally, we state our conclusions in SectionV.

II. PHYSICS IN FU SE D MACH IN E LEARNING ARCHITECTURES

Fig.1 shows the two proposed PIML frameworks as well as the vanilla low-ﬁdelity physics model (L). In this paper, all the

models shown are used to predict the airframe force and moment coefﬁcients for a given set of control and environmental

parameters. The PIML models amalgamate low-ﬁdelity physics models with Artiﬁcial Neural Networks (ANNs) to enhance

predictions, yet they vary based on the speciﬁc role of the ANNs. As depicted in the framework diagram 1, the low-ﬁdelity

model comprises of a propeller solver and the Vortex Lattice Method (VLM) model. The prop solver computes induced

velocities which are then fed into the VLM, which in turn calculates the force and moment coefﬁcients on the airframe.

The PIML-A model incorporates ANN layers to achieve three primary objectives: 1) To supersede the prop solver and

directly estimate the axial and tangential induced velocities of the propellers, 2) to modify the ﬂight conditions in order to

enhance the predictions of the VLM model, and 3) to adjust the output of the VLM to reduce discrepancies with respect to the

high-ﬁdelity data. Given that all layers are trained concurrently, the PIML-A model learns the mapping from ﬂight conditions

2

Induced

Veloci�es

VLM Model

Prop Solver

Inputs

Inputs

Low-Fidelity Outputs

Low-Fidelity Model

(a)

PIML -A

Transfer Layers

VLM Model

PIML Predicted Output

Transfer Parameters Correc�on Layers

Inputs

(b)

Prop Solver

Inputs

PIML -B

Induced

Veloci�es

VLM Model

PIML Predicted Output

Correc�on Layers

Inputs

Predicted Correc�on

of Induced Veloci�es

(c)

ANN Predicted Output

Inputs

Pure Ar�ﬁcial Neural Network

(d)

Fig. 1: Framework Diagrams of a) Low-Fidelity Physics b) PIML-A c) PIML-B and d) Purely Data-Driven ANN

and control inputs to the transfer parameters. Simultaneously, it learns the output errors of the VLM and adjusts them to align

more closely with the high-ﬁdelity data. The reasoning behind this model is that part of the errors in the VLM model can be

mitigated by shifting the inputs into different domains, and the errors that remain, not resolved by this input modiﬁcation, will

be addressed by the correction layers on the output side.

In contrast, the PIML-B model does not adjust any inputs to the original low-ﬁdelity model and retains the prop solver in its

structure. Within this design, the sole objective is to identify the error in the induced velocities computed by the prop solver

and rectify these parameters. This model is more closely aligned with the low-ﬁdelity physics model and aims to enhance

modeling accuracy by reﬁning the outputs of the prop solver.

The following subsection focuses on the training processes of these PIML models, speciﬁcally the computation of gradients

and the implementation of backpropagation through the low-ﬁdelity physics models.

3

A. PIML Model Training

The training of neural networks (i.e optimizing the weights of the neural network to minimize loss) is performed using

gradient descent and therefore requires the gradient of the loss function(L) w.r.t to the weights(W). The following equations

show the integration of backpropagation gradients through an auto-differentiable physics model. This feature enables the training

of our PIML architectures using standard ANN training techniques and using standard ANN software libraries.

L=∥Yi−Y′

i∥2(MSE Loss)

L=1

n

n

i=1

(Yi−Y′

i)2

∴

∂L

∂W =1

n

n

i=0

2 (Y′

i−Yi)∂Y ′

i

∂W

Where Yi

Ground Truth

=ϕ(Xi),Y′

i

Prediction

=G(Xi, W )

(1)

Grepresents the sequential PIML model and ϕrepresents the high-ﬁdelity physics model.The terms Xi,Yi,Y′

irepresent the

input, high-ﬁdelity output and the OPTMA prediction of the ith training sample respectively.

From the deﬁnition of Y′

i:

Y′

i=G(Xi, W )

∂Y ′

i

∂W =∂ G (Xi, W )

∂W

As Gis the OPTMA model, we have:

G(Xi, W ) = Φ (S(Xi, W ))

=⇒∂G(Xi, W )

∂W =∂Φ (S(Xi, W ))

∂W =∂Φ(U)

∂U

Partial Physics

Auto-Differentiation

×∂S(Xi, W )

∂W

Backpropagation

(2)

Here, Sand Φrepresent the neural network and the partial physics models respectively. Urepresents the intermediate parameter

vector which is the output of the transfer network and input to the partial physics model.

III. CAS E STU DY: MODELLING AERODYNAM IC S OF A TILT-ROTOR E VTOL AIRCRAFT

The Aerobeacon, shown in ﬁg.2, is a 6 rotor aircraft with 4 hover rotors in a quad conﬁguration and two tilt-rotors at the

edges of the wings. The aircraft has no rudder and instead makes use of differential thrust between the tilt rotors for yaw

control. The only movable control surface on the aircraft apart from the tiltrotor is an elevator. Since this is our initial attempt

at modelling this aircraft, we chose to ﬁx the hover rotor RPMs at 5000 and only vary the other control inputs listed in Tale.I.

Table I: Aircraft Control Parameter Bounds

Parameters Bounds

Speed (v) [0 m/s ,45 m/s]

Angle of Attack (α) [-15◦, 15◦]

Starboard Propeller

RPM (ωstar)

[4000 , 10000]

Port Propeller RPM

(ωport)

[4000 , 10000]

Starboard Propeller

Angle (θstar)

[0◦, 110◦]

Port Propeller Angle

(θport)

[0◦, 110◦]

Elevator Deﬂection

(θelev)

[-15◦, 15◦]

Fig. 2: Aerobeacon aircraft design conﬁguration from Flighthouse

Engineering

4

A. Modelling Aircraft Flight Dynamics

Table II displays the inputs and outputs of all the models and also the transfer parameters corresponding to the PIML-A

model. The transfer parameters are speciﬁcally selected such that they have an impact on the output accuracy of the model, as

well as to allow greater ﬂexibility to the embedded ANN layers to affect the ﬁnal outputs. In both the proposed architectures,

we try to improve the low-ﬁdelity model by either augmenting or replacing the prop solver. The PIML-A model in particular

also tries to alter the outputs of the VLM model to improve accuracy. This model is an extension of the OPTMA architecture

discussed in the Introduction.

Table II: Modeling Parameters

Inputs (All Models) Transfer Parameters (PIML-A Model) Outputs (All Models)

Speed (v)

Angle of Attack (α)

Starboard Propeller RPM

(ωstar)

Port Propeller RPM (ωport)

Starboard Propeller Angle

(θstar)

Port Propeller Angle (θport)

Elevator Deﬂection (θelev)

Speed(v′)

Angle of Attack(α′)

Elevator Deﬂection(θ′

elev)

−→

va

star

−→

vt

star

−→

va

port

−→

vt

port

−→

va

hover

−→

vt

hover

CL

CD

Cl

Cm

B. Low Fidelity Model

The lower ﬁdelity ﬂight dynamics model uses Bechamo LLC’s developed VLM plus propeller effects low-ﬁdelity tool,

BLOFI. This tool was developed initially to model the NASA Tiltwing aircraft and was validated against the X-57 data [41].

The validation plots can be seen in ﬁg.9. This was done in order to prove that we are able to capture the blown wing effects.

BLOFI was created by using an improvised VLM that accounts for some blown wing effects caused by the propellers upstream

of the the lifting surfaces. A base VLM [42] is implemented and is modiﬁed by accounting for induced velocities caused by

propellers’ wash upstream of the control points on the lifting surfaces. The properties for each propeller are calculated using

Blade Element Momentum Theory [43]. It should also be noted that propeller slipstream contraction [44] is accounted for

when including the induced velocities from the propellers in the VLM setup.

A set of ﬂight conditions is created by making a grid of every combination of freestream velocity, sideslip and angle of

attack. At each of these ﬂight conditions the aerodynamic forces and moments, including the propeller thrusts and torques,

are calculated, using the aforementioned VLM implementation. This is done for every expected RPM setting, propeller tilt

angle and tail deﬂection. The combination of aerodynamic forces and moments and propeller thrusts and torques is used to

determine the overall forces and moments acting on the aircraft.

CHARM[45] is a commercial tool developed by Continuum Dynamics, Inc. for simulating the aerodynamics and dynamics

of rotorcraft using partical methods. Flighthouse Engineering LLC. provided their Aerobeacon aircraft design, parameters,

and constructed the CHARM model. They ran CHARM test cases for a collection of randomly generated input samples. The

data set derived from the CHARM model serves as the high-ﬁdelity data for the research presented in this paper. The VLM

conﬁguration is shown in ﬁg.3 and the CHARM higher ﬁdelity output visualization is shown in ﬁg.4.

C. Sparse High-Fidelity Dataset

In order to generate the high ﬁdelity dataset, we used Latin Hypercube Sampling and created a dataset of 100 points across

the parameters and bounds listed in Tab.I. These 100 samples were then input to the CHARM high ﬁdelity model and the

dataset was generated. Out of the 100 initial samples, only 87 were usable as 11 cases failed to converge on the CHARM

model and the remaining 2 cases showcased abnormally high values of the force and moment coefﬁcients and were removed

from the dataset. While generating the dataset, we ﬁxed the RPMs of the hover quad rotors to 5000 RPM. This ﬁnal dataset of

87 samples was then randomly divided into a training set containing 70 samples and a validation set containing 17 samples.

All the performance results shown in this paper are run on this validation dataset consisting of samples completely unseen by

the model.

D. Prior Analysis of BLOFI Performance on Other Data-Sets

BLOFI was developed at Bechamo LLC and in order to prove its usefulness in comparison to other available VLM solvers

that do not account for blown wing effects. It has been previously validated using CFD data[41] generated by NASA for the

X-57. Just as a ﬂat plate model was created for the Aerobeacon with the appropriate propellers for the developments of this

paper, a ﬂat plate model with the corresponding propellers was earlier created for the X-57 as part of the validation study

5

Table III: Comparison of PIML, Low-Fidelity and Pure ANN Models

PIML-A PIML-B Low-Fidelity ANN

Execution Time (s) 0.32 0.53 0.338 0.0002

Training Time (min) 23.46 23.18 - 0.04

#Hidden Layers 6 6 - 4

#Nodes per Layer 200 200 - 150

on BLOFI. In ﬁgure9 we are able to see that BLOFI can capture signiﬁcant blown-wing effects, by comparing the power-on

and power-off plots. The increase in CLand CDat every angle of attack can be observed. More importantly, the inclusion

of induced velocities in the VLM solve also lets us observe the increased ﬂap effectiveness. Errors are between 5%and 10%

for the range of angles of attack for which this validation study was conducted. It is expected that with a PIML approach,

not only can we close down this gap (error), but more readily extend the low/medium ﬁdelity solver such as BLOFI to work

on a wider variety of aircraft conﬁgurations with little to no manual tuning and manual model development. To support his

premise, the next section presents the results of our PIML architectures with the example of Aerobeacon conﬁguration, where

we had more control over the high-ﬁdelity sampling – enabling ease of PIML training for this paper.

IV. RES ULT S AN D DISCUSSION

A. Baselines For Comparison

For testing the performance of the two proposed PIML models, we compared them against the vanilla low-ﬁdelity model(L)

shown in ﬁg.1a and a pure data-driven ANN (A). Table III lists the parameters of the stand-alone ANN and the ANNs used as

part of the PIML frameworks. All the data-driven and PIML models were trained on the same training dataset of 70 high-ﬁdelity

samples and validated on a validation dataset consisting of 17 samples. As the training data was sparse, a pure ANN with

similar size of the ANNs used for the PIML models tended to overﬁt very quickly. Therefore, the size of the pure data-driven

ANN is lower as compared to the ANNs used in the PIML models.

B. Modelling Performance

Figure.5 shows the convergence history for both the PIML models. From the ﬁgure we see that PIML-A has a much smoother

training history as compared to the PIML-B model. The PIML-B model seems to be sensitive to corrections in induced velocity.

Further training with a lower learning rate or decaying the learning rate may resolve this issue and improve performance of

the PIML-B model. It may also be the case that the VLM is very sensitive to the induced velocity inputs or that there is an

unknown bias in the validation dataset. This could’ve happened due to the random splitting of data into train and test samples.

Further work with varying test and train datasets similar to K-fold cross-validation are necessary to understand if there’s any

data bias.

Figure.6 shows the prediction error of the models in all of the 4 outputs. From the ﬁgure, we see that the PIML-A model

performs the best. The pure data-driven ANN and the PIML-B model exhibit similar performance. Relative to the amount

Fig. 3: Mesh model of the Aerobeacon aircraft. The

propellers and their direction of rotation are represented

by directed circles. The two rotors positioned at the wing

extremities are capable of tilting.

Fig. 4: CHARM visualized output of particle method to

obtain a higher-ﬁdelity model at discrete points

6

0 20 40 60 80 100 120 140

0

0.05

0.1

0.15

0.2

Train Loss

Test Loss

Epoch

MSE Loss

(a) PIML-A

0 20 40 60 80 100 120 140

0

0.05

0.1

0.15

0.2

Train Loss

Test Loss

Epoch

MSE Loss

(b) PIML-B

Fig. 5: Convergence history of PIML Models

Low_Fidelity PIML A PIML B ANN

0

0.2

0.4

0.6

0.8

1

M

o

d

e

l

RMSE (Normalized)

(a) Prediction Error in CL

Low_Fidelity PIML A PIML B ANN

0

0.2

0.4

0.6

0.8

1

M

o

d

e

l

RMSE (Normalized)

(b) Prediction Error in CD

Low_Fidelity PIML A PIML B ANN

0

0.2

0.4

0.6

0.8

1

M

o

d

e

l

RMSE (Normalized)

(c) Prediction Error in Cl

Low_Fidelity PIML A PIML B ANN

0

0.2

0.4

0.6

0.8

1

M

o

d

e

l

RMSE (Normalized)

(d) Prediction Error in Cm

Fig. 6: Prediction Error Compared to High Fidelity Data

of data available, the ANN performs very well and can prove to be a possible alternative in cases with a larger sample set

and no additional requirements on interpretability. he fact that the high-ﬁdelity samples were generated using Latin hypercube

sampling, ensuring an even domain coverage, could also be a contributing factor to the ANN’s strong performance. Among

all the modeled outputs, PIML-A appears to excel in predicting CD. This could be attributed to the transfer parameters having

a greater impact on CDas compared to the other outputs. Where things begin to get interesting is when we see the effect of

the learnt output correction layers in PIML-A as shown in ﬁg.7. From the ﬁgure, it is easily seen that the output layers of

the VLM model have a very strong negative correction on the CLthat the VLM outputs. As we also see from ﬁg.6, we see

that the VLM model has the highest prediction error in CL. Combined, these two plots convey that the transfer parameters

7

C_L C_D C_l C_m

−1.5

−1

−0.5

0

0.5

1

Model

Normalized Correction to VLM Outputs

Fig. 7: Correction of VLM Outputs by the Correction Layers

in PIML-A

PIML-B

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Model

Normalized Correction to Induced Velocities

Fig. 8: Correction of Propeller Induced Velocities by the

Correction Layers in PIML-B

do not inﬂuence CLas much as they do the other outputs or that just shifting the inputs is not sufﬁcient to improve the CL

predictions of the VLM model. The correction for other outputs predicted by the PIML model is evenly distributed between

positive and negative and this seems to be reasonable given the performance of the vanilla low-ﬁdelity models on all these

outputs is about the same.

The PIML-B model on the other hand performs on par with the low-ﬁdelity VLM model. This outcome might be explained

by an earlier hypothesis suggesting that if the induced velocities indeed have a minor impact on CL, then regardless of what

the correction network learns, it won’t be able to enhance the predictions for CL. However, since the training loss is calculated

across all outputs, the overall loss remains substantial, causing the network to struggle in learning the transfer mapping necessary

to improve predictions on the other inputs. It may be the case that when different models are trained to learn the various outputs

of this modelling problem we might see improved performance by the PIML-B architecture on outputs other than CL, but as

that is not feasible especially once the number of outputs gets large enough to support design / control frameworks, PIML-A

model might be better suited for this problem.

V. CONCLUSION

The paper presents two distinct architectures of physics-informed models, referred to as PIML-A and PIML-B. These models

integrate Artiﬁcial Neural Networks with a low-ﬁdelity BLOFI model with the aim of enhancing the prediction performance

of the aerodynamic coefﬁcients of a six-rotor eVTOL aircraft. To facilitate the training of both models, an end-to-end auto-

differentiable physics framework was established using Google JAX. The models were then trained using a sparse dataset

derived from a high-ﬁdelity CHARM model. When comparing prediction performance, it was found that the PIML-A model

achieved a higher degree of accuracy in comparison to the PIML-B model, the basic low-ﬁdelity model, and a baseline purely

data-driven model. A comprehensive analysis of the results demonstrated that the PIML-A model’s method of simultaneous

input-shifting and output-correction yielded superior results, given that one of the inputs might not be highly sensitive to the

chosen transfer parameters. To conclusively determine the optimal architecture for this modeling problem, additional testing

using cross-validation and hyperparameter optimization is needed. However, the initial results indicate that the PIML-A model

holds signiﬁcant promise. Looking forward, the BLOFI low-ﬁdelity model, when augmented with PIML, could potentially be

employed to generalize across diverse aircraft designs and conﬁgurations, thereby proving to be a valuable tool in the domains

of aircraft design and control.

ACK NOW LE DG EM EN TS

This material is based upon work funded by Bechamo LLC’s (with sub-contract to University at Buffalo) NASA Phase II

Small Business Innovation Research (SBIR) Award No. 80NSSC22CA046. The authors would also like to thank Flighthouse

8

Engineering LLC for providing the aircraft design and the high-ﬁdelity data used in this paper.

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APPENDIX

Fig. 9: BLOFI was veriﬁed against X-57 CFD data.

11