How to structure integral constraints？

[kecbh]

The question is -u''=λu, λ is 1, and 0<x<pi, and u(0)=u(pi)=0.\int_0^{\pi} u(x)dx=\pi,I want to implement integral constraints, and there is currently no solution instability for integral constraint training.
Code as follow,

import deepxde as dde
import numpy as np
import matplotlib.pyplot as plt

geom = dde.geometry.Interval(0, np.pi)


def pde(x, y):
    dy_xx = dde.grad.hessian(y, x)
    return -dy_xx - y

def func(x):
    return np.sin(x) 

data = dde.data.PDE(geom, pde, [], num_domain=64, solution=func, num_test=400)

layer_size = [1] + [50] * 3 + [1]
activation = 'tanh'
initializer = 'Glorot uniform'
net = dde.nn.FNN(layer_size, activation, initializer)

def output_transform(x, y):
    return x *(np.pi-x) * y + x

net.apply_output_transform(output_transform)

model = dde.Model(data, net)
model.compile("adam", lr=1e-4, metrics=["l2 relative error"])

losshistory, train_state = model.train(iterations=5000)

dde.saveplot(losshistory, train_state, issave=True, isplot=True)

I want to know how to add integral constraints to the code to guarantee the stability of the solution. I know you can use model.compile, but how do I do that? I don't know how deepxde is implemented.

[lululxvi]

Use OperatorBC. Check https://doi.org/10.1137/21M1397908 and code.

[yogibadepalli]

Dear @kecbh,

I'm currently working on implementing an integral constraint to normalize the output of a neural network, specifically using the following condition:

(\int_{0}^{\infty} |r^2| , dr = 1)

To achieve this, I need to incorporate the integral constraint into the training process (deepxde). I noticed from your earlier query that you were exploring a similar approach. I wanted to reach out and ask if you’ve found a solution or workaround for implementing this constraint effectively.

I don't think the functionality of operatorBC helps in adding integral constraint in the implementation.

Any insights or suggestions you can share would be greatly appreciated.

Best Regards,
Yogeswara.