Neural network based solvers for partial differential equations and inverse problems
P. Stiller, F. Bethke, M. Böhme, R. Pausch, S. Torge, A. Debus, J. Vorberger, M.Bussmann, N. Hoffmann:
Large-scale Neural Solvers for Partial Differential Equations (2020).
Raissi, Maziar, Paris Perdikaris, and George Em Karniadakis.
Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations.(2017).
Raissi, Maziar, Paris Perdikaris, and George Em Karniadakis.
Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations.(2017).
cuda 10.2 # if gpu support is needed
python/3.6.5
gcc/5.5.0
openmpi/3.1.2
torch==1.7.1
h5py==2.10.0
numpy==1.19.0
Pillow==7.2.0
matplotlib==3.3.3
scipy==1.6.1
pyDOE==0.3.8
At the beginning you have to implement the datasets following the torch.utils.Dataset interface
from torch.utils.data import Dataset
sys.path.append(PATH_TO_PINN_FRAMEWORK) # adding the pinn framework to your path
import PINNFramework as pf
class BoundaryConditionDataset(Dataset):
def __init__(self, nb, lb, ub):
"""
Constructor of the initial condition dataset
"""
def __getitem__(self, idx):
"""
Returns data for initial state
"""
def __len__(self):
"""
Length of the dataset
"""
class InitialConditionDataset(Dataset):
def __init__(self, **kwargs):
"""
Constructor of the boundary condition dataset
"""
def __len__(self):
"""
Length of the dataset
"""
def __getitem__(self, idx):
"""
Returns item at given index
"""
class PDEDataset(Dataset):
def __init__(self, nf, lb, ub):
"""
Constructor of the PDE dataset
"""
def __len__(self):
"""
Length of the dataset
"""
def __getitem__(self, idx):
"""
Returns item at given index
"""
In the main function you can create the loss-terms and the corresponding datasets. And define the pde function f which is the residual of the pde given residual points and model predictions u. For the boundary conditions: neumann, robin, dirchlet and periodic boundary condititions are supported.
if __name__ == main :
# initial condition
ic_dataset = InitialConditionDataset(...)
initial_condition = pf.InitialCondition(dataset=ic_dataset)
# boundary conditions
bc_dataset = BoundaryConditionDataset(...)
periodic_bc_u = pf.PeriodicBC(...)
periodic_bc_v = pf.PeriodicBC(...)
periodic_bc_u_x = pf.RobinBC(...)
periodic_bc_v_x = pf.NeumannBC(...)
# PDE
pde_dataset = PDEDataset(...)
def f(x, u):
"""
Defines the residual of the pde f(x,u)=0
"""
pde_loss = pf.PDELoss(dataset=pde_dataset, func=f)Finally you can create a model which is the surrogate for the PDE and create the PINN enviorment which helps you to train the surrogate.
model = pf.models.MLP(input_size=2, output_size=2, hidden_size=100, num_hidden=4) # creating a model. For example a mlp
pinn = pf.PINN(model, input_size=2, output_size=2 ,pde_loss = pde_loss, initial_condition=initial_condition, boundary_condition = [...], use_gpu=True)
pinn.fit(50000, 'Adam', 1e-3)Instead of a PDE loss you can use a HPM model. The HPM model needs a function derivatives that calculates the needed derivatives, while the last returned derivative is the time_derivative. You can use the HPM loss a follows.
der derivatives(x,u):
"""
Returns the derivatives
Args:
x (torch.Tensor) : residual points
u (torch.Tensor) : predictions of the pde model
"""
pass
hpm_loss = pf.HPMLoss(pde_dataset,derivatives,hpm_model)
#HPM has no boundary conditions in general
pinn = pf.PINN(model, input_size=2, output_size=2 ,pde_loss = hpm_loss, initial_condition=initial_condition, boundary_condition = [], use_gpu=True)You can activate horovod support by setting the use_horovod flag in the constructor of the pinn
pinn = pf.PINN(model, input_size=2, output_size=2 ,pde_loss = pde_loss, initial_condition=initial_condition, boundary_condition = [...], use_gpu=True, use_horovod=True)Keep in mind that the lbfgs-optimizer and the lbgfgs-finetuning is not supported with horovod activated. Another restriction is that the length or your dataset should not be smaller than the number of used GPUs for horovod.