# -*- coding: utf-8 -*-importnumpyasnpimportmath# Create random input and output datax=np.linspace(-math.pi,math.pi,2000)y=np.sin(x)# Randomly initialize weightsa=np.random.randn()b=np.random.randn()c=np.random.randn()d=np.random.randn()learning_rate=1e-6fortinrange(2000):# Forward pass: compute predicted y# y = a + b x + c x^2 + d x^3y_pred=a+b*x+c*x**2+d*x**3# Compute and print lossloss=np.square(y_pred-y).sum()ift%100==99:print(t,loss)# Backprop to compute gradients of a, b, c, d with respect to lossgrad_y_pred=2.0*(y_pred-y)grad_a=grad_y_pred.sum()grad_b=(grad_y_pred*x).sum()grad_c=(grad_y_pred*x**2).sum()grad_d=(grad_y_pred*x**3).sum()# Update weightsa-=learning_rate*grad_ab-=learning_rate*grad_bc-=learning_rate*grad_cd-=learning_rate*grad_dprint(f'Result: y ={a} +{b} x +{c} x^2 +{d} x^3')

# -*- coding: utf-8 -*-importtorchimportmathdtype=torch.floatdevice=torch.device("cpu")# device = torch.device("cuda:0") # Uncomment this to run on GPU# Create random input and output datax=torch.linspace(-math.pi,math.pi,2000,device=device,dtype=dtype)y=torch.sin(x)# Randomly initialize weightsa=torch.randn((),device=device,dtype=dtype)b=torch.randn((),device=device,dtype=dtype)c=torch.randn((),device=device,dtype=dtype)d=torch.randn((),device=device,dtype=dtype)learning_rate=1e-6fortinrange(2000):# Forward pass: compute predicted yy_pred=a+b*x+c*x**2+d*x**3# Compute and print lossloss=(y_pred-y).pow(2).sum().item()ift%100==99:print(t,loss)# Backprop to compute gradients of a, b, c, d with respect to lossgrad_y_pred=2.0*(y_pred-y)grad_a=grad_y_pred.sum()grad_b=(grad_y_pred*x).sum()grad_c=(grad_y_pred*x**2).sum()grad_d=(grad_y_pred*x**3).sum()# Update weights using gradient descenta-=learning_rate*grad_ab-=learning_rate*grad_bc-=learning_rate*grad_cd-=learning_rate*grad_dprint(f'Result: y ={a.item()} +{b.item()} x +{c.item()} x^2 +{d.item()} x^3')

importtorchimportmath# We want to be able to train our model on an `accelerator <https://pytorch.org/docs/stable/torch.html#accelerators>`__# such as CUDA, MPS, MTIA, or XPU. If the current accelerator is available, we will use it. Otherwise, we use the CPU.dtype=torch.floatdevice=torch.accelerator.current_accelerator().typeiftorch.accelerator.is_available()else"cpu"print(f"Using{device} device")torch.set_default_device(device)# Create Tensors to hold input and outputs.# By default, requires_grad=False, which indicates that we do not need to# compute gradients with respect to these Tensors during the backward pass.x=torch.linspace(-1,1,2000,dtype=dtype)y=torch.exp(x)# A Taylor expansion would be 1 + x + (1/2) x**2 + (1/3!) x**3 + ...# Create random Tensors for weights. For a third order polynomial, we need# 4 weights: y = a + b x + c x^2 + d x^3# Setting requires_grad=True indicates that we want to compute gradients with# respect to these Tensors during the backward pass.a=torch.randn((),dtype=dtype,requires_grad=True)b=torch.randn((),dtype=dtype,requires_grad=True)c=torch.randn((),dtype=dtype,requires_grad=True)d=torch.randn((),dtype=dtype,requires_grad=True)initial_loss=1.learning_rate=1e-5fortinrange(5000):# Forward pass: compute predicted y using operations on Tensors.y_pred=a+b*x+c*x**2+d*x**3# Compute and print loss using operations on Tensors.# Now loss is a Tensor of shape (1,)# loss.item() gets the scalar value held in the loss.loss=(y_pred-y).pow(2).sum()# Calculare initial loss, so we can report loss relative to itift==0:initial_loss=loss.item()ift%100==99:print(f'Iteration t ={t:4d}  loss(t)/loss(0) ={round(loss.item()/initial_loss,6):10.6f}  a ={a.item():10.6f}  b ={b.item():10.6f}  c ={c.item():10.6f}  d ={d.item():10.6f}')# Use autograd to compute the backward pass. This call will compute the# gradient of loss with respect to all Tensors with requires_grad=True.# After this call a.grad, b.grad. c.grad and d.grad will be Tensors holding# the gradient of the loss with respect to a, b, c, d respectively.loss.backward()# Manually update weights using gradient descent. Wrap in torch.no_grad()# because weights have requires_grad=True, but we don't need to track this# in autograd.withtorch.no_grad():a-=learning_rate*a.gradb-=learning_rate*b.gradc-=learning_rate*c.gradd-=learning_rate*d.grad# Manually zero the gradients after updating weightsa.grad=Noneb.grad=Nonec.grad=Noned.grad=Noneprint(f'Result: y ={a.item()} +{b.item()} x +{c.item()} x^2 +{d.item()} x^3')

importtorchimportmathclassLegendrePolynomial3(torch.autograd.Function):"""    We can implement our own custom autograd Functions by subclassing    torch.autograd.Function and implementing the forward and backward passes    which operate on Tensors.    """@staticmethoddefforward(ctx,input):"""        In the forward pass we receive a Tensor containing the input and return        a Tensor containing the output. ctx is a context object that can be used        to stash information for backward computation. You can cache tensors for        use in the backward pass using the ``ctx.save_for_backward`` method. Other        objects can be stored directly as attributes on the ctx object, such as        ``ctx.my_object = my_object``. Check out `Extending torch.autograd <https://docs.pytorch.org/docs/stable/notes/extending.html#extending-torch-autograd>`_        for further details.        """ctx.save_for_backward(input)return0.5*(5*input**3-3*input)@staticmethoddefbackward(ctx,grad_output):"""        In the backward pass we receive a Tensor containing the gradient of the loss        with respect to the output, and we need to compute the gradient of the loss        with respect to the input.        """input,=ctx.saved_tensorsreturngrad_output*1.5*(5*input**2-1)dtype=torch.floatdevice=torch.device("cpu")# device = torch.device("cuda:0")  # Uncomment this to run on GPU# Create Tensors to hold input and outputs.# By default, requires_grad=False, which indicates that we do not need to# compute gradients with respect to these Tensors during the backward pass.x=torch.linspace(-math.pi,math.pi,2000,device=device,dtype=dtype)y=torch.sin(x)# Create random Tensors for weights. For this example, we need# 4 weights: y = a + b * P3(c + d * x), these weights need to be initialized# not too far from the correct result to ensure convergence.# Setting requires_grad=True indicates that we want to compute gradients with# respect to these Tensors during the backward pass.a=torch.full((),0.0,device=device,dtype=dtype,requires_grad=True)b=torch.full((),-1.0,device=device,dtype=dtype,requires_grad=True)c=torch.full((),0.0,device=device,dtype=dtype,requires_grad=True)d=torch.full((),0.3,device=device,dtype=dtype,requires_grad=True)learning_rate=5e-6fortinrange(2000):# To apply our Function, we use Function.apply method. We alias this as 'P3'.P3=LegendrePolynomial3.apply# Forward pass: compute predicted y using operations; we compute# P3 using our custom autograd operation.y_pred=a+b*P3(c+d*x)# Compute and print lossloss=(y_pred-y).pow(2).sum()ift%100==99:print(t,loss.item())# Use autograd to compute the backward pass.loss.backward()# Update weights using gradient descentwithtorch.no_grad():a-=learning_rate*a.gradb-=learning_rate*b.gradc-=learning_rate*c.gradd-=learning_rate*d.grad# Manually zero the gradients after updating weightsa.grad=Noneb.grad=Nonec.grad=Noned.grad=Noneprint(f'Result: y ={a.item()} +{b.item()} * P3({c.item()} +{d.item()} x)')

# -*- coding: utf-8 -*-importtorchimportmath# Create Tensors to hold input and outputs.x=torch.linspace(-math.pi,math.pi,2000)y=torch.sin(x)# For this example, the output y is a linear function of (x, x^2, x^3), so# we can consider it as a linear layer neural network. Let's prepare the# tensor (x, x^2, x^3).p=torch.tensor([1,2,3])xx=x.unsqueeze(-1).pow(p)# In the above code, x.unsqueeze(-1) has shape (2000, 1), and p has shape# (3,), for this case, broadcasting semantics will apply to obtain a tensor# of shape (2000, 3)# Use the nn package to define our model as a sequence of layers. nn.Sequential# is a Module which contains other Modules, and applies them in sequence to# produce its output. The Linear Module computes output from input using a# linear function, and holds internal Tensors for its weight and bias.# The Flatten layer flatens the output of the linear layer to a 1D tensor,# to match the shape of `y`.model=torch.nn.Sequential(torch.nn.Linear(3,1),torch.nn.Flatten(0,1))# The nn package also contains definitions of popular loss functions; in this# case we will use Mean Squared Error (MSE) as our loss function.loss_fn=torch.nn.MSELoss(reduction='sum')learning_rate=1e-6fortinrange(2000):# Forward pass: compute predicted y by passing x to the model. Module objects# override the __call__ operator so you can call them like functions. When# doing so you pass a Tensor of input data to the Module and it produces# a Tensor of output data.y_pred=model(xx)# Compute and print loss. We pass Tensors containing the predicted and true# values of y, and the loss function returns a Tensor containing the# loss.loss=loss_fn(y_pred,y)ift%100==99:print(t,loss.item())# Zero the gradients before running the backward pass.model.zero_grad()# Backward pass: compute gradient of the loss with respect to all the learnable# parameters of the model. Internally, the parameters of each Module are stored# in Tensors with requires_grad=True, so this call will compute gradients for# all learnable parameters in the model.loss.backward()# Update the weights using gradient descent. Each parameter is a Tensor, so# we can access its gradients like we did before.withtorch.no_grad():forparaminmodel.parameters():param-=learning_rate*param.grad# You can access the first layer of `model` like accessing the first item of a listlinear_layer=model[0]# For linear layer, its parameters are stored as `weight` and `bias`.print(f'Result: y ={linear_layer.bias.item()} +{linear_layer.weight[:,0].item()} x +{linear_layer.weight[:,1].item()} x^2 +{linear_layer.weight[:,2].item()} x^3')

# -*- coding: utf-8 -*-importtorchimportmath# Create Tensors to hold input and outputs.x=torch.linspace(-math.pi,math.pi,2000)y=torch.sin(x)# Prepare the input tensor (x, x^2, x^3).p=torch.tensor([1,2,3])xx=x.unsqueeze(-1).pow(p)# Use the nn package to define our model and loss function.model=torch.nn.Sequential(torch.nn.Linear(3,1),torch.nn.Flatten(0,1))loss_fn=torch.nn.MSELoss(reduction='sum')# Use the optim package to define an Optimizer that will update the weights of# the model for us. Here we will use RMSprop; the optim package contains many other# optimization algorithms. The first argument to the RMSprop constructor tells the# optimizer which Tensors it should update.learning_rate=1e-3optimizer=torch.optim.RMSprop(model.parameters(),lr=learning_rate)fortinrange(2000):# Forward pass: compute predicted y by passing x to the model.y_pred=model(xx)# Compute and print loss.loss=loss_fn(y_pred,y)ift%100==99:print(t,loss.item())# Before the backward pass, use the optimizer object to zero all of the# gradients for the variables it will update (which are the learnable# weights of the model). This is because by default, gradients are# accumulated in buffers( i.e, not overwritten) whenever .backward()# is called. Checkout docs of torch.autograd.backward for more details.optimizer.zero_grad()# Backward pass: compute gradient of the loss with respect to model# parametersloss.backward()# Calling the step function on an Optimizer makes an update to its# parametersoptimizer.step()linear_layer=model[0]print(f'Result: y ={linear_layer.bias.item()} +{linear_layer.weight[:,0].item()} x +{linear_layer.weight[:,1].item()} x^2 +{linear_layer.weight[:,2].item()} x^3')

# -*- coding: utf-8 -*-importtorchimportmathclassPolynomial3(torch.nn.Module):def__init__(self):"""        In the constructor we instantiate four parameters and assign them as        member parameters.        """super().__init__()self.a=torch.nn.Parameter(torch.randn(()))self.b=torch.nn.Parameter(torch.randn(()))self.c=torch.nn.Parameter(torch.randn(()))self.d=torch.nn.Parameter(torch.randn(()))defforward(self,x):"""        In the forward function we accept a Tensor of input data and we must return        a Tensor of output data. We can use Modules defined in the constructor as        well as arbitrary operators on Tensors.        """returnself.a+self.b*x+self.c*x**2+self.d*x**3defstring(self):"""        Just like any class in Python, you can also define custom method on PyTorch modules        """returnf'y ={self.a.item()} +{self.b.item()} x +{self.c.item()} x^2 +{self.d.item()} x^3'# Create Tensors to hold input and outputs.x=torch.linspace(-math.pi,math.pi,2000)y=torch.sin(x)# Construct our model by instantiating the class defined abovemodel=Polynomial3()# Construct our loss function and an Optimizer. The call to model.parameters()# in the SGD constructor will contain the learnable parameters (defined# with torch.nn.Parameter) which are members of the model.criterion=torch.nn.MSELoss(reduction='sum')optimizer=torch.optim.SGD(model.parameters(),lr=1e-6)fortinrange(2000):# Forward pass: Compute predicted y by passing x to the modely_pred=model(x)# Compute and print lossloss=criterion(y_pred,y)ift%100==99:print(t,loss.item())# Zero gradients, perform a backward pass, and update the weights.optimizer.zero_grad()loss.backward()optimizer.step()print(f'Result:{model.string()}')

# -*- coding: utf-8 -*-importrandomimporttorchimportmathclassDynamicNet(torch.nn.Module):def__init__(self):"""        In the constructor we instantiate five parameters and assign them as members.        """super().__init__()self.a=torch.nn.Parameter(torch.randn(()))self.b=torch.nn.Parameter(torch.randn(()))self.c=torch.nn.Parameter(torch.randn(()))self.d=torch.nn.Parameter(torch.randn(()))self.e=torch.nn.Parameter(torch.randn(()))defforward(self,x):"""        For the forward pass of the model, we randomly choose either 4, 5        and reuse the e parameter to compute the contribution of these orders.        Since each forward pass builds a dynamic computation graph, we can use normal        Python control-flow operators like loops or conditional statements when        defining the forward pass of the model.        Here we also see that it is perfectly safe to reuse the same parameter many        times when defining a computational graph.        """y=self.a+self.b*x+self.c*x**2+self.d*x**3forexpinrange(4,random.randint(4,6)):y=y+self.e*x**expreturnydefstring(self):"""        Just like any class in Python, you can also define custom method on PyTorch modules        """returnf'y ={self.a.item()} +{self.b.item()} x +{self.c.item()} x^2 +{self.d.item()} x^3 +{self.e.item()} x^4 ? +{self.e.item()} x^5 ?'# Create Tensors to hold input and outputs.x=torch.linspace(-math.pi,math.pi,2000)y=torch.sin(x)# Construct our model by instantiating the class defined abovemodel=DynamicNet()# Construct our loss function and an Optimizer. Training this strange model with# vanilla stochastic gradient descent is tough, so we use momentumcriterion=torch.nn.MSELoss(reduction='sum')optimizer=torch.optim.SGD(model.parameters(),lr=1e-8,momentum=0.9)fortinrange(30000):# Forward pass: Compute predicted y by passing x to the modely_pred=model(x)# Compute and print lossloss=criterion(y_pred,y)ift%2000==1999:print(t,loss.item())# Zero gradients, perform a backward pass, and update the weights.optimizer.zero_grad()loss.backward()optimizer.step()print(f'Result:{model.string()}')