# Neural Net not able to learn simple analytical equation

I am currently making my first attempts with Pytorch. I am trying to solve a simple equation with a neural net. Analytically solved, the result of my neural net shall look like this: $$y = \frac{x_5}{x_2} - \frac{x_1-x_2}{2 x_3 x_4}\frac{x_2}{x_1}$$ while I am randomly generating input data with the following boundaries: $$500 \leq x_1 \leq 1000 \\ 1 \leq x_2 \leq x_1 \\ 1 \leq x_5 \leq 10000 \\ 1e-6\leq x_4 \leq 1e-3 \\ 1000 \leq x_3 \leq 50000$$ Now the neural network does not seem to be able to learn the function. I assume this is due to the large spreads of the input areas of the function. I have already tried it with various network architectures and different activation functions. I have also tried an lr scheduler and varied the learning rate from 1e-2-1e-8. I have also tried to summarize the input variables $$x_3x_4$$ as a separate variable.

Since I would like to add additional equations to that neural network to be solved later on, I implemented the following, normalized, loss function:

F.l1_loss((y_est+(x_1-x_2)/(2*x_3*x_4)*x_2/x_1 - x_5/x_2)/(x_5/x_2),torch.FloatTensor(np.zeros(batch_size)))


I mainly worked here with nn of multiple sizes and layers, but couldnt get good results (mainly losses over 0.5, but bouncing up to 4ish)

Edit:

import torch.optim as optim
import numpy as np
import torch
import torch.nn as nn
import torch.nn.functional as F
import random
from torch.optim.lr_scheduler import ReduceLROnPlateau

def get_batch(batch_size=32):
batch_x = []
for i in range(batch_size):
x_1 = random.uniform(500,1000)
x_2 = random.uniform(1,x_1-1)
x_3 = random.uniform(1,10000)
x_4 = random.uniform(1e-6,0.001)
x_5 = np.random.randint(1000,50000)
batch_x.append([x_1,x_2,x_3,x_4,x_5])

class CustomNet(nn.Module):
def __init__(self,n_input,n_output,n_hidden_neurons,n_hidden_layers):
super(CustomNet, self).__init__()
self.sequential_layers = nn.Sequential(
*((nn.Linear(n_input,n_hidden_neurons),)+ tuple(nn.Linear(n_hidden_neurons,n_hidden_neurons) for i in range(n_hidden_layers)))
)
self.fc1 = nn.Linear(n_hidden_neurons,n_output)

def forward(self, x):
x = torch.log(x)
output = self.sequential_layers(x)
output = torch.exp(output)
output = self.fc1(output)
return output

batch_size = 1000
n_epochs = 50000
n_input,n_output,n_hidden_neurons,n_hidden_layers = 5,1,16,1

# create model
nnet = CustomNet(n_input,n_output,n_hidden_neurons,n_hidden_layers)
nnet = nnet.to("cuda")

lr, counter = 0.01, 0
target = torch.FloatTensor(np.zeros(batch_size)).contiguous().cuda()
lr_scheduler = ReduceLROnPlateau(optimizer, mode='min', factor=0.1, patience=1000, verbose=True)
target = torch.FloatTensor(np.zeros(batch_size)).cuda().contiguous()
for i in range(n_epochs):

batch_x = get_batch(batch_size)
x_1,x_2,x_3,x_4,x_5 = batch_x.T
y_est = nnet(batch_x)
y_est = y_est[:,0]
output = F.l1_loss((y_est+(x_1-x_2)/(2*x_3*x_4)*x_2/x_1 - x_5/x_2)/(x_5/x_2),target)

# Backward pass
loss = output.item()
output.backward()
optimizer.step()
lr_scheduler.step(output)

• Have you tried first overfitting on just some training samples? Additionally, it would help if you could provide a full code example (i.e. generating the training data as well as creating and training the model) as this would help us better diagnose any possible issues. Nov 17, 2023 at 18:33
• I've updated the question with a code example. Thanks for your help. Nov 17, 2023 at 19:03

• add nonlinearity between hidden layers and increase the number of hidden layers: currently self.sequential_layers is just a stacked linear transformation. Add ReLU between consecutive hidden layer should help you learn quicker;
• not sure if torch.log and torch.exp is a good idea. You may remove the torch.log and replace torch.exp with your choice of nonlinearity for bridging the hidden layers;