By H. Martin Bücker, George Corliss, Paul Hovland, Uwe Naumann, Boyana Norris
This assortment covers the state-of-the-art in automated differentiation concept and perform. Practitioners and scholars will know about advances in computerized differentiation options and methods for the implementation of sturdy and strong instruments. Computational scientists and engineers will enjoy the dialogue of functions, which supply perception into powerful concepts for utilizing automated differentiation for layout optimization, sensitivity research, and uncertainty quantification.
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Extra info for Automatic Differentiation: Applications, Theory, and Implementations
22 Paul J. Werbos In actuality, the challenge of supervised learning was not what really brought me to develop backpropagation. That was a later development. My initial goal was to develop a kind of universal neural network learning device to perform a kind of “Reinforcement Learning” (RL) illustrated in Fig. 5. External Environment or "Plant" U(t) X(t) sensor inputs "utility" or "reward" or "reinforcement" RLS u(t) actions Fig. 5. A Concept of Reinforcement Learning. The environment and the RLS are both assumed to have memory at time t of the previous time t − 1.
We have: x(t) P (T, X, Y ) (arcsin t)/t (1 − T 2 )(X + T Y )2 − 1 t/(arcsin t) Explicit Rational ODE 2 None 2 (X − T Y ) (1 − T ) − X 2 4 None (arctan t)/t (X + T Y )(1 + T ) − 1 x˙ = [1 − x(1 + t2 )]/[t(1 + t2 )] t/(arctan t) (X − T Y )(1 + T 2 ) − X 2 x˙ = [x(1 + t2 ) − x2 ]/[t(1 + t2 )]. Proof. One will suﬃce, say x = (arctan t)/t. We set s = sin(tx) , c = cos(tx) . 42 Harley Flanders Then tx = arctan t, t = tan(tx), s = tc and s˙ = (x + ty)c, c˙ = −(x + ty)s . Diﬀerentiate s = tc: (x + ty)c = c − t(x + ty)s, x + ty = 1 − t2 (x + ty), (x + ty)(1 + t2 ) = 1 .
At that time, I focused on the case where the utility function U depends only on the state x, and not on the current actions u. I discussed how the reverse calculations could be implemented in a local way, in a distributed system of computing hardware like the brain. Harvard responded as follows to this proposal and to later discussions. First, they would not allow ANNs as such to be a major part of the thesis, since I had not found anyone willing to act as a mentor for that part. D. D. Third, they had some skepticism about reverse diﬀerentiation itself, and they wanted a really solid, clear, rigorous proof of its validity in the general case.
Automatic Differentiation: Applications, Theory, and Implementations by H. Martin Bücker, George Corliss, Paul Hovland, Uwe Naumann, Boyana Norris