3 ms·

> I think you missed the point. Navier-Stokes does not explain Lift, nor does Bernoulli's law or Netwon's third. Like the friction under an ice skate, it's a ph

by shoyer 7y ago

> I think you missed the point. Navier-Stokes does not explain Lift, nor does Bernoulli's law or Netwon's third. Like the friction under an ice skate, it's a phenomenon we take for granted in science and engineering without a rigorous explanation In physics and engineering, the challenge is to make mathematical models of the world that accurately predict how it works. Once we've done that we can declare success. I agree with Yann that it is not useful to focus on "intuitive" explanations and a fixation on qualitative casual reasoning is largely useless, e.g., very few physicists care about the "why" of quantum theory. But there is a big difference between how we can explain how a plane flies and how we can explain how a neural network works: - The airplane flying doesn't have a "simple" explanation but it can be explained based on fundamental physical principles and mathematics. These models seem to have nearly unlimited ability to extrapolate (within their known bounds of validity). - The neural network giving accurate predictions can only (for the most part) be explained empirically. The models fail to extrapolate and worse, most of the time we can't even trust the model to know when it's extrapolating. > But you can, and not just conceivably, but practically train a neural net to approximate navier Stokes to a high degree of accuracy with a fraction of the computation of a 3D finite model. This is a lovely idea, but I as someone currently doing research in this space (deep learning for CFD) I don't think its true. There are loads of neural nets that can accurately approximate Navier-Stokes within some limited domain of applicability (e.g., for flow over a parameterized set of airfoils), but so far nothing is close to matching the generalization power of Navier-Stokes.

srean 7y ago
> In physics and engineering, the challenge is to make mathematical models of the world that accurately predict how it works. Once we've done that we can declare success. With enough epicycles one can make the heliocentric model as accurate as one wants as far as forecasting positions of planets. A system of DNN sequence models will do just as good, if not better. Since we regard Einstein highly (at least more highly than an undergrad who can train a TensorFlow model on planetary position data), it makes me think there is more to physics than having a mathematical model with good predictive accuracy.
- shoyer 7y ago
  It's actually extraordinarily difficult to get a DNN model to accurately predict planetary motion over arbitrary time horizons. RNNs have a tendency to either decay or blow-up. The only way I know to make neural nets work well for this problem is to build in lots of physics into the model architecture, e.g., conservation of energy: https://arxiv.org/abs/1906.01563 https://arxiv.org/abs/1906.01563

by shoyer 7y ago

srean 7y ago

> In physics and engineering, the challenge is to make mathematical models of the world that accurately predict how it works. Once we've done that we can declare success. With enough epicycles one can make the heliocentric model as accurate as one wants as far as forecasting positions of planets. A system of DNN sequence models will do just as good, if not better. Since we regard Einstein highly (at least more highly than an undergrad who can train a TensorFlow model on planetary position data), it makes me think there is more to physics than having a mathematical model with good predictive accuracy.

shoyer 7y ago
It's actually extraordinarily difficult to get a DNN model to accurately predict planetary motion over arbitrary time horizons. RNNs have a tendency to either decay or blow-up. The only way I know to make neural nets work well for this problem is to build in lots of physics into the model architecture, e.g., conservation of energy: https://arxiv.org/abs/1906.01563 https://arxiv.org/abs/1906.01563