Answers.Body: <p>It has. Gradient flow or more generally <a href="https://en.wikipedia.org/wiki/Flow_(mathematics)#Time-dependent_ordinary_differential_equations" rel="nofollow noreferrer">flow</a> is a well known concept in maths. Say we have a function <span class="math-container">$f:\mathbb R^n \longrightarrow \mathbb R^n$</span> and a function <span class="math-container">$\theta:[0,\infty)\longrightarrow \mathbb R^n$</span> such that the ODE
  <span class="math-container">$$
  \partial_t \theta(t) = f(\theta(t))
  $$</span>
  exists for any initial choice <span class="math-container">$\theta(0)\in\mathbb R^n$</span> uniquely. Then i think the naming comes from the following informal description. Lets imagine the case <span class="math-container">$n=2$</span> then <span class="math-container">$\mathbb R^2$</span> is a plane. Now you drop <span class="math-container">$\theta(0)$</span> somewhere on this plane and wait some time <span class="math-container">$t$</span> to see where the <span class="math-container">$\theta(0)$</span> will <em>flow</em> to. You can also draw the trajectories for multiple starting points <span class="math-container">$\theta(0)$</span> which will often give you an image that looks like a flow of a liquid.</p>
  <p>In the case of the ANN <span class="math-container">$f$</span> would be the negative gradient of the costfunction with respect to the paramter-vector <span class="math-container">$\theta$</span> and gradient descent would be an approximation of the gradient flow.</p>
  
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Answers.Body: <p>Here is my idea of what that means: Gradient flow is an abstract term to describe properties of the gradient. The gradient is calculated by propagating the error backwards through the networks, therefore it kind of <em>flows</em> from the last to the first layer. Depending on network architecture and loss function the flow can behave differently.</p>
  <p>One popular kind of undesirable gradient flow is the <strong>vanishing gradient</strong>. It refers to the gradient norm being very small, i.e. the parameter updates are very small which slows down/prevents proper training. It often occurs when training very deep neural networks. Residual connections can help, because they bypass operations that reduce gradient magnitude.</p>
  <p>The exact opposite would be <strong>exploding gradients</strong> where your gradient-norm is very large which leads to unstable training and the weights inside the network can't follow a stable trajectory to the optimum. This often occurs in the context of recurrent neural networks.</p>
  <p><a href="https://www.analyticsvidhya.com/blog/2021/06/the-challenge-of-vanishing-exploding-gradients-in-deep-neural-networks/" rel="nofollow noreferrer">Here</a> is a nice article on vanishing/exploding gradients.</p>
  <p>A general way to phrase this is that the gradient reveals properties about the loss function which you are optimizing (as it is the derivative). <strong>A noisy or heavily oscillating loss function usually implies an undesirable gradient flow</strong>.</p>
  <p>Thinking about these properties plays an important role in network and loss function design.
  For example, the root function <span class="math-container">$\sqrt{\cdot}$</span> has a very large derivative when approaching zero, so having a <span class="math-container">$\sqrt{\cdot}$</span> in the network or the loss function can lead to exploding gradients.</p>
  
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Body: <p>Several research papers and textbooks (e.g. <a href="https://www.mdpi.com/2227-9709/8/3/53" rel="nofollow noreferrer">this</a>) contain the phrase &quot;<strong>gradient flow</strong>&quot; in the context of neural networks.</p>
  <p>I am confused about whether it has any rigorous and formal way of understanding or not. What is the flow referring to here?</p>
  
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Comments.Text: Note that although this is seemingly not the way it's used in the paper you link, gradient flows in the context of optimization generally refer to a continuous-time analogue of gradient descent; see, e.g., here: https://francisbach.com/gradient-flows/
Comments.UserDisplayName: Hypercube
Tags: <deep-learning><terminology><math><definitions><gradient>
Title: What does it mean by "gradient flow" in the context of neural networks?
UserDisplayName: hanugm
ThreadText: # What does it mean by "gradient flow" in the context of neural networks?
  
  (Asked by: username_0 on 2022-03-16)
  
  Several research papers and textbooks (e.g. [this](https://www.mdpi.com/2227-9709/8/3/53)) contain the phrase "**gradient flow**" in the context of neural networks.
  
  I am confused about whether it has any rigorous and formal way of understanding or not. What is the flow referring to here?
  
  **Tags:** deep-learning, terminology, math, definitions, gradient
  
  > **Comments:**
  > **username_1**: Note that although this is seemingly not the way it's used in the paper you link, gradient flows in the context of optimization generally refer to a continuous-time analogue of gradient descent; see, e.g., here: https://francisbach.com/gradient-flows/  
  
  ---
  
  ## 2 answers
  
  ### Answer by username_2 on 2022-07-22. Score: 4
  
  It has. Gradient flow or more generally [flow](https://en.wikipedia.org/wiki/Flow_(mathematics)#Time-dependent_ordinary_differential_equations) is a well known concept in maths. Say we have a function $f:\mathbb R^n \longrightarrow \mathbb R^n$ and a function $\theta:[0,\infty)\longrightarrow \mathbb R^n$ such that the ODE
  $$
  \partial\_t \theta(t) = f(\theta(t))
  $$
  exists for any initial choice $\theta(0)\in\mathbb R^n$ uniquely. Then i think the naming comes from the following informal description. Lets imagine the case $n=2$ then $\mathbb R^2$ is a plane. Now you drop $\theta(0)$ somewhere on this plane and wait some time $t$ to see where the $\theta(0)$ will *flow* to. You can also draw the trajectories for multiple starting points $\theta(0)$ which will often give you an image that looks like a flow of a liquid.
  
  In the case of the ANN $f$ would be the negative gradient of the costfunction with respect to the paramter-vector $\theta$ and gradient descent would be an approximation of the gradient flow.
  
  
  ---
  
  ### Answer by username_3 on 2022-03-16. Score: 1
  
  Here is my idea of what that means: Gradient flow is an abstract term to describe properties of the gradient. The gradient is calculated by propagating the error backwards through the networks, therefore it kind of *flows* from the last to the first layer. Depending on network architecture and loss function the flow can behave differently.
  
  One popular kind of undesirable gradient flow is the **vanishing gradient**. It refers to the gradient norm being very small, i.e. the parameter updates are very small which slows down/prevents proper training. It often occurs when training very deep neural networks. Residual connections can help, because they bypass operations that reduce gradient magnitude.
  
  The exact opposite would be **exploding gradients** where your gradient-norm is very large which leads to unstable training and the weights inside the network can't follow a stable trajectory to the optimum. This often occurs in the context of recurrent neural networks.
  
  [Here](https://www.analyticsvidhya.com/blog/2021/06/the-challenge-of-vanishing-exploding-gradients-in-deep-neural-networks/) is a nice article on vanishing/exploding gradients.
  
  A general way to phrase this is that the gradient reveals properties about the loss function which you are optimizing (as it is the derivative). **A noisy or heavily oscillating loss function usually implies an undesirable gradient flow**.
  
  Thinking about these properties plays an important role in network and loss function design.
  For example, the root function $\sqrt{\cdot}$ has a very large derivative when approaching zero, so having a $\sqrt{\cdot}$ in the network or the loss function can lead to exploding gradients.
  
  
  ---
  
  