Adaptive design and optimization using physics-informed neural networks

1 . A system for adaptive design and optimization of an engineering design using a physics-informed neural network, the system comprising:

a non-transitory memory storing instructions; and

a processor in communication with the non-transitory memory, wherein, the processor executes the instructions to cause the system to:

input collocation points and design parameters into the physics-informed neural network to obtain an output;

calculate a loss function based on a set of governing equations and the output;

determine whether the physics-informed neural network is convergent based on the calculated loss function;

in response to the physics-informed neural network being convergent, export the physics-informed neural network; and

in response to the physics-informed neural network not being convergent:

determine whether to resample the collocation points;

determine an optimum number of collocation points;

determine a set of optimal network parameters for adjusting the physics-informed neural network, wherein the set of optimal network parameters are determined by integrating a physics-informed automatic simulation point grid adaptation (PI-AutoGrid) into the physics-informed neural network to calculate an importance metric at each pre-specified grid point corresponding to each of the collocation points; and

input the collocation points and the set of optimal network parameters to the physics-informed neural network for a next iteration.

2 . The system according to claim 1 , wherein when the processor executes the instructions to cause the system to determine the optimum number of collocation points, the processor executes the instructions to cause the system to:

determine the optimum number of collocation points based on at least one of the following:

a convergence rate based on a boundary condition (BC);

a convergence rate based on an initial condition (IC);

a convergence rate based on a partial differential equations (PDE);

a user-defined output variable fidelity requirement;

a gradient minimum;

a gradient maximum;

a resource constraint; or

an adjoint loss function.

3 . The system according to claim 1 , wherein when the processor executes the instructions to cause the system to determine the optimum number of collocation points, the processor executes the instructions to cause the system to:

calculate a set of metrics;

normalize the set of metrics; and

redistribute the collocation points.

4 . The system according to claim 3 , wherein when the processor executes the instructions to cause the system to normalize the set of metrics, the processor executes the instructions to cause the system to:

normalize the set of metrics based on a resource constraint, wherein after the normalization, the optimum number of collocation points is within the resource constraint.

5 . The system according to claim 1 , wherein:

the set of governing equations comprises a set of gradient computing equations and a set of loss computing equations; and

when the processor executes the instructions to cause the system to calculate the loss function based on the set of governing equations, the processor executes the instructions to cause the system to:

calculate a set of gradients based on the set of gradient computing equations; and

calculate the loss function based on the set of gradients and the set of loss computing equations.

6 . The system according to claim 5 , wherein:

the set of loss computing equations comprises at least one of the following:

a partial differential equation;

a boundary condition (BC) equation;

an initial condition (IC) equation; or

an adjoint loss function.

7 . The system according to claim 1 , wherein:

the set of optimal network parameters includes at least one of the following: a network architecture, a set of activation functions, a set of normalization layers, a set of loss weightings, or a set of learning rate schedules.

8 . The system according to claim 1 , wherein when the processor executes the instructions to cause the system to determine the set of optimal network parameters, the processor executes the instructions to cause the system to:

determine a set of optimal network parameters based on at least one of a context constraint and a resource constraint.

9 . The system according to claim 1 , wherein when the processor executes the instructions to cause the system to determine the set of optimal network parameters, the processor executes the instructions to cause the system to:

determine the set of optimal network parameters based on at least one of the following:

a problem geometry;

a parametrization range;

a problem physics;

a requested output fidelity;

a requested output field; or

an available resource.

10 . A method for adaptive design and optimization of an engineering design using a physics-informed neural network, the method comprising:

inputting, by a device comprising a memory storing instructions and a processor in communication with the memory, collocation points and design parameters into the physics-informed neural network to obtain an output;

calculating, by the device, a loss function based on a set of governing equations and the output;

determining, by the device, whether the physics-informed neural network is convergent based on the calculated loss function;

in response to the physics-informed neural network being convergent, exporting, by the device, the physics-informed neural network; and

in response to the physics-informed neural network not being convergent:

determining, by the device, whether to resample the collocation points;

determining, by the device, an optimum number of collocation points;

determining, by the device, a set of optimal network parameters for adjusting the physics-informed neural network, wherein the set of optimal network parameters are determined by integrating a physics-informed automatic simulation point grid adaptation (PI-AutoGrid) into the physics-informed neural network to calculate an importance metric for each network parameter at every pre-specified grid point corresponding to each of the collocation points; and

inputting, by the device, the collocation points and the set of optimal network parameters to the physics-informed neural network for a next iteration.

11 . The method according to claim 10 , wherein the determining the optimum number of collocation points comprises:

determining, by the device, the optimum number of collocation points based on at least one of the following:

a convergence rate based on a boundary condition (BC);

a convergence rate based on an initial condition (IC);

a convergence rate based on a partial differential equations (PDE);

a user-defined output variable fidelity requirement;

a gradient minimum;

a gradient maximum;

a resource constraint; or

an adjoint loss function.

12 . The method according to claim 10 , wherein the determining the optimum number of collocation points comprises:

calculating, by the device, a set of metrics;

normalizing, by the device, the set of metrics; and

redistributing, by the device, the collocation points.

13 . The method according to claim 12 , wherein the normalizing the set of metrics comprises:

normalizing, by the device, the set of metrics based on a resource constraint, wherein after the normalization, the optimum number of collocation points is within the resource constraint.

14 . The method according to claim 10 , wherein:

the set of governing equations comprises a set of gradient computing equations and a set of loss computing equations; and

the calculating the loss function based on the set of governing equations comprises:

calculating, by the device, a set of gradients based on the set of gradient computing equations; and

calculating, by the device, the loss function based on the set of gradients and the set of loss computing equations.

15 . The method according to claim 14 , wherein:

the set of loss computing equations comprises at least one of the following:

a partial differential equation;

a boundary condition (BC) equation;

an initial condition (IC) equation; or

an adjoint loss function.

16 . The method according to claim 10 , wherein:

the set of optimal network parameters includes at least one of the following: a network architecture, a set of activation functions, a set of normalization layers, a set of loss weightings, or a set of learning rate schedules.

17 . The method according to claim 10 , wherein the determining the set of optimal network parameters comprises:

determining, by the device, a set of optimal network parameters based on at least one of a context constraint and a resource constraint.

18 . The method according to claim 10 , wherein the determining the set of optimal network parameters comprises:

determining, by the device, the set of optimal network parameters based on at least one of the following:

a problem geometry;

a parametrization range;

a problem physics;

a requested output fidelity;

a requested output field; or

an available resource.

19 . A product comprising:

machine-readable media other than a transitory signal;

instructions stored on the machine-readable media; and

wherein when a processor executes the instructions, the product is configured to cause the processor to:

input collocation points and design parameters into a physics-informed neural network to obtain an output;

calculate a loss function based on a set of governing equations and the output;

determine whether the physics-informed neural network is convergent based on the calculated loss function;

in response to the physics-informed neural network being convergent, export the physics-informed neural network; and

in response to the physics-informed neural network not being convergent:

determine whether to resample the collocation points;

determine an optimum number of collocation points;

determine a set of optimal network parameters for adjusting the physics-informed neural network, wherein the set of optimal network parameters are determined by integrating a physics-informed automatic simulation point grid adaptation (PI-AutoGrid) into the physics-informed neural network to calculate an importance metric for each network parameter at every pre-specified grid point corresponding to each of the collocation points; and

input the collocation points and the set of optimal network parameters to the physics-informed neural network for a next iteration.

20 . The product according to claim 19 , wherein when the product is configured to cause the processor to determine the optimum number of collocation points, the product is configured to cause the processor to:

determine the optimum number of collocation points based on at least one of the following:

a convergence rate based on a boundary condition (BC);

a convergence rate based on an initial condition (IC);

a convergence rate based on a partial differential equations (PDE);

a user-defined output variable fidelity requirement;

a gradient minimum;

a gradient maximum;

a resource constraint; or

an adjoint loss function.