Physics-informed neural network for inversely predicting effective material properties of metamaterials
Conventionally, design methodologies employed deep learning model based on physics which considers only real (permittivity) term and ignores the imaginary (conductivity) term in complex loss function which fail to help in design of complex structures and limit their applications to scenarios such as array of metamaterial structures. Present application provides systems and method implement apply a Physics-Informed Neural Network (PINN) for inversely calculating the effective material parameters of a multi-dimensional metamaterial from its scattered field(s). By employing a loss function based on the Helmholtz wave equation, performance of a metamaterial is modeled by the system the dependance of resonant behavior on the homogenized electric permittivity distribution profile generated by the PINN is demonstrated.
1 . A processor implemented method for physics-informed neural network (PINN) for inversely predicting effective material properties of metamaterials, comprising:
obtaining, via one or more hardware processors, an input comprising a plurality of surface coordinates corresponding to a multi-dimensional metamaterial with meta-atom designs of a disc and a split-ring resonator (SRR);
performing, via the one or more hardware processors, a finite element method-based wave simulation (FEMWS) on the plurality of surface coordinates associated with the multi-dimensional metamaterial to obtain a corresponding electric field profile;
segregating, via the one or more hardware processors, the plurality of surface coordinates associated with the multi-dimensional metamaterial and the corresponding electric field profile into a first region and a second region to obtain a first region-based data and a second region-based data, wherein the first region is a ring region of the multi-dimensional metamaterial and the second region is a non-ring region of the multi-dimensional metamaterial;
obtaining, via the one or more hardware processors, one or more material properties comprising (i) an electric permittivity, (ii) an electric conductivity, and a wave number for each of the first region-based data and the second region-based data;
normalizing, via the one or more hardware processors, the electric permittivity, the electric conductivity, the first region-based data, and the second region-based data to obtain a first set of normalized data and a second set of normalized data;
creating, via the one or more hardware processors, the Physics-Informed Neural Network (PINN) based on the first set of normalized data and the second set of normalized data for homogenizing the disc and the SRR;
computing, via the one or more hardware processors, a loss function corresponding to the created PINN for each of the first region and the second region using the first set of normalized data and the second set of normalized data and the wave number, wherein the loss function for the first region is computed using an expression:
∇
×
∇
×
E
(
x
,
y
)
+
k
0
2
ε
r
(
x
,
y
)
E
(
x
,
y
)
-
jk
0
σ
(
x
,
y
)
E
(
x
,
y
)
=
0
,
and wherein E is the corresponding electric field profile, x and y are the plurality of surface coordinates, k 0 is the wave number, ε r is the electric permittivity, σ is the electric conductivity, and j is a complex conjugate, wherein when three coordinates are obtained for the multi-dimensional metamaterial, then the expression is re-written to incorporate the z coordinate as:
∇
×
∇
×
E
(
x
,
y
,
z
)
+
k
0
2
ε
r
(
x
,
y
,
z
)
E
(
x
,
y
,
z
)
=
0
,
wherein the loss function for the second region is computed using an expression:
∇
×
∇
×
E
(
x
,
y
)
+
k
0
2
ε
r
(
x
,
y
)
E
(
x
,
y
)
=
0
and wherein E is the corresponding electric field profile, x and y are the plurality of surface coordinates, k 0 is the wave number, ε r is the electric permittivity, and σ is the electric conductivity for the plurality of surface coordinates x and y, wherein when three coordinates are obtained for the multi-dimensional metamaterial, then the expression is re-written to incorporate the z coordinate as:
∇
×
∇
×
E
(
x
,
y
,
z
)
+
k
0
2
ε
r
(
x
,
y
,
z
)
E
(
x
,
y
,
z
)
=
0
;
iteratively training, via the one or more hardware processors, the PINN using the loss function computed for each of the first region and the second region until an optimized PINN is obtained, wherein the optimized PINN comprises an optimized loss function;
applying, via the optimized PINN, the optimized loss function on the plurality of surface coordinates of the multi-dimensional metamaterial to predict an electric field profile using associated weights and bias of the optimized PINN;
inversely calculating, via the one or more hardware processors, one or more effective material properties at the plurality of surface coordinates for the first region and the second region using the predicted electric field profile and a Helmholtz equation;
computing a first average effective material property based on the one or more effective material properties calculated for the ring region and the non-ring region;
simulating, via the finite element method-based wave simulation (FEMWS), the first average effective material property to obtain a first simulated electric field profile;
performing a comparison of (i) the first simulated electric field profile, and (ii) the corresponding electric field profile;
validating the first simulated electric field profile based on the comparison;
computing a second average of effective material property based on the one or more effective material properties calculated for the ring region and the non-ring region;
feeding back the second average of effective material property to a software tool for validating an output generated by the optimized PINN; and
re-simulating the first average effective material property with the effective material property and obtaining a second simulated electric field profile for a resonant frequency.
2 . The processor implemented method of claim 1 , wherein the first region and the second region are different from each other.
3 . The processor implemented method of claim 1 , wherein the plurality of surface coordinates obtained from the first set of normalized data and the second set of normalized data serves as one or more inputs corresponding to one or more input layers of the PINN, and wherein the corresponding electric field from the first set of normalized data and the second set of normalized data serves as an output corresponding to an output layer of the PINN.
4 . The processor implemented method of claim 1 , wherein the one or more effective material properties comprise at least one of an effective electric permittivity and an effective electric conductivity.
5 . A system for physics-informed neural network for inversely predicting effective material properties of metamaterials, comprising:
a memory storing instructions;
one or more communication interfaces; and
one or more hardware processors coupled to the memory via the one or more communication interfaces, wherein the one or more hardware processors are configured by the instructions to:
obtain an input comprising a plurality of surface coordinates corresponding to a multi-dimensional metamaterial with meta-atom designs of a disc and a split-ring resonator (SRR);
perform a finite element method-based wave simulation (FEMWS) on the plurality of surface coordinates associated with the multi-dimensional metamaterial to obtain a corresponding electric field profile;
segregate the plurality of surface coordinates associated with the multi-dimensional metamaterial and the corresponding simulated electric field profile into a first region and a second region to obtain a first region-based data and a second region-based data, wherein the first region is a ring region of the multi-dimensional metamaterial and the second region is a non-ring region of the multi-dimensional metamaterial;
obtain one or more material properties comprising (i) an electric permittivity, (ii) an electric conductivity, and a wave number for each of the first region-based data and the second region-based data;
normalize the electric permittivity, the electric conductivity, the first region-based data, and the second region-based data to obtain a first set of normalized data and a second set of normalized data;
create the Physics-Informed Neural Network (PINN) based on the first set of normalized data and the second set of normalized data for homogenizing the disc and the SRR;
compute a loss function corresponding to the created PINN for each of the first region and the second region using the first set of normalized data and a second set of normalized data and the wave number, wherein the loss function for the first region is computed using an expression:
∇
×
∇
×
E
(
x
,
y
)
+
k
0
2
ε
r
(
x
,
y
)
E
(
x
,
y
)
-
jk
0
σ
(
x
,
y
)
E
(
x
,
y
)
=
0
,
and wherein E is the corresponding electric field profile, x and y are the plurality of surface coordinates, k 0 is the wave number, ε r is the electric permittivity, σ is the electric conductivity, and j is a complex conjugate, wherein when three coordinates are obtained for the multi-dimensional metamaterial, then the expression is re-written to incorporate the z coordinate as:
∇
×
∇
×
E
(
x
,
y
,
z
)
+
k
0
2
ε
r
(
x
,
y
,
z
)
E
(
x
,
y
,
z
)
=
0
,
wherein the loss function for the second region is computed using an expression:
∇
×
∇
×
E
(
x
,
y
)
+
k
0
2
ε
r
(
x
,
y
)
E
(
x
,
y
)
=
0
and wherein E is the corresponding electric field profile, x and y are the plurality of surface coordinates, k 0 is the wave number, ε r is the electric permittivity, and σ is the electric conductivity for the plurality of surface coordinates x and y, wherein when three coordinates are obtained for the multi-dimensional metamaterial, then the expression is re-written to incorporate the z coordinate as:
∇
×
∇
×
E
(
x
,
y
,
z
)
+
k
0
2
ε
r
(
x
,
y
,
z
)
E
(
x
,
y
,
z
)
=
0
;
iteratively train the PINN using the loss function computed for each of the first region and the second region until an optimized PINN is obtained, wherein the optimized PINN comprises an optimized loss function;
apply, via the optimized PINN, the optimized loss function on the plurality of surface coordinates of the multi-dimensional metamaterial to predict an electric field profile using associated weights and bias of the optimized PINN;
inversely calculate one or more effective material properties at the plurality of surface coordinates for the first region and the second region using the predicted electric field profile and a Helmholtz equation;
compute a first average effective material property based on the one or more effective material properties calculated for the ring region and the non-ring region;
simulate, via the finite element method-based wave simulation (FEMWS), the first average effective material property to obtain a first simulated electric field profile;
perform a comparison of (i) the first simulated electric field profile, and (ii) the corresponding electric field profile;
validate the first simulated electric field profile based on the comparison;
compute a second average of effective material property based on the one or more effective material properties calculated for the ring region and the non-ring region;
feedback the second average of effective material property to a software tool for validating an output generated by the optimized PINN; and
re-simulate the first average effective material property with the effective material property and obtain a second simulated electric field profile for a resonant frequency.
6 . The system of claim 5 , wherein the first region and the second region are different from each other.
7 . The system of claim 5 , wherein the plurality of surface coordinates obtained from the first set of normalized data and the second set of normalized data serves as one or more inputs corresponding to one or more input layers of the PINN, and wherein the corresponding electric field from the first set of normalized data and the second set of normalized data serves as an output corresponding to an output layer of the PINN.
8 . The system of claim 5 , wherein the one or more effective material properties comprise at least one of an effective electric permittivity and an effective electric conductivity.
9 . One or more non-transitory machine-readable information storage mediums comprising one or more instructions which when executed by one or more hardware processors cause:
obtaining an input comprising a plurality of surface coordinates corresponding to a multi-dimensional metamaterial with meta-atom designs of a disc and a split-ring resonator (SRR);
performing a finite element method-based wave simulation (FEMWS) on the plurality of surface coordinates associated with the multi-dimensional metamaterial to obtain a corresponding electric field profile;
segregating the plurality of surface coordinates associated with the multi-dimensional metamaterial and the corresponding electric field profile into a first region and a second region to obtain a first region-based data and a second region-based data, wherein the first region is a ring region of the multi-dimensional metamaterial and the second region is a non-ring region of the multi-dimensional metamaterial;
obtaining one or more material properties comprising (i) an electric permittivity, (ii) an electric conductivity, and a wave number for each of the first region-based data and the second region-based data;
normalizing the electric permittivity, the electric conductivity, the first region-based data, and the second region-based data to obtain a first set of normalized data and a second set of normalized data;
creating the Physics-Informed Neural Network (PINN) based on the first set of normalized data and the second set of normalized data for homogenizing the disc and the SRR;
computing a loss function corresponding to the created PINN for each of the first region and the second region using the first set of normalized data and the second set of normalized data and the wave number, wherein the loss function for the first region is computed using an expression:
∇
×
∇
×
E
(
x
,
y
)
+
k
0
2
ε
r
(
x
,
y
)
E
(
x
,
y
)
-
jk
0
σ
(
x
,
y
)
E
(
x
,
y
)
=
0
,
and wherein E is the corresponding electric field profile, x and y are the plurality of surface coordinates, k 0 is the wave number, ε r is the electric permittivity, σ is the electric conductivity, and j is a complex conjugate, wherein when three coordinates are obtained for the multi-dimensional metamaterial, then the expression is re-written to incorporate the z coordinate as:
∇
×
∇
×
E
(
x
,
y
,
z
)
+
k
0
2
ε
r
(
x
,
y
,
z
)
E
(
x
,
y
,
z
)
=
0
,
wherein the loss function for the second region is computed using an expression:
∇
×
∇
×
E
(
x
,
y
)
+
k
0
2
ε
r
(
x
,
y
)
E
(
x
,
y
)
=
0
and wherein E is the corresponding electric field profile, x and y are the plurality of surface coordinates, k 0 is the wave number, ε r is the electric permittivity, and σ is the electric conductivity for the plurality of surface coordinates x and y, wherein when three coordinates are obtained for the multi-dimensional metamaterial, then the expression is re-written to incorporate the z coordinate as:
∇
×
∇
×
E
(
x
,
y
,
z
)
+
k
0
2
ε
r
(
x
,
y
,
z
)
E
(
x
,
y
,
z
)
=
0
;
iteratively training the PINN using the loss function computed for each of the first region and the second region until an optimized PINN is obtained, wherein the optimized PINN comprises an optimized loss function;
applying, via the optimized PINN, the optimized loss function on the plurality of surface coordinates of the multi-dimensional metamaterial to predict an electric field profile using associated weights and bias of the optimized PINN; and
inversely calculating one or more effective material properties at the plurality of surface coordinates for the first region and the second region using the predicted electric field profile and a Helmholtz equation;
computing a first average effective material property based on the one or more effective material properties calculated for the ring region and the non-ring region;
simulating, via the finite element method-based wave simulation (FEMWS), the first average effective material property to obtain a first simulated electric field profile;
performing a comparison of (i) the first simulated electric field profile, and (ii) the corresponding electric field profile;
validating the first simulated electric field profile based on the comparison;
computing a second average of effective material property based on the one or more effective material properties calculated for the ring region and the non-ring region;
feeding back the second average of effective material property to a software tool for validating an output generated by the optimized PINN; and
re-simulating the first average effective material property with the effective material property and obtaining a second simulated electric field profile for a resonant frequency.
10 . The one or more non-transitory machine-readable information storage mediums of claim 9 , wherein the first region and the second region are different from each other.
11 . The one or more non-transitory machine-readable information storage mediums of claim 9 , wherein the plurality of surface coordinates obtained from the first set of normalized data and the second set of normalized data serves as one or more inputs corresponding to one or more input layers of the PINN, and wherein the corresponding electric field from the first set of normalized data and the second set of normalized data serves as an output corresponding to an output layer of the PINN.
12 . The one or more non-transitory machine-readable information storage mediums of claim 9 , wherein the one or more effective material properties comprise at least one of an effective electric permittivity and an effective electric conductivity.