Gradient Descent for High Dimensional Systems

Transcription

1 Gradient Descent for High Dimensional Systems

2 Lab versus Lab 2 D Geometry Op>miza>on Poten>al Energy Methods: Implemented Equa3ons for op3mizer Bond length High Dimensional Op>miza>on Applica3ons: Used so;ware to op3mize high dimensional systems; perform calcula3ons to predict proper3es of materials Today we will fill in the math we lei out of how gradient descent will work for high dimensional systems. AIer lecture, you will apply the content to a worksheet.

3 D Poten>al Energy Surface (PES) Diatomic Molecule (H 2, O 2, etc.) Two Noble Atoms (Ar 2, etc.) Poten>al Energy, V Bond length, r Energy required to break a bond Poten>al Energy, V Bond length, r Op>mal Bondlength The majority of chemical systems will need more than one dimension to describe its structure

4 2D Poten>al Energy Surface (PES) We can view 2D PES by u>lizing contour plots. Contour plots contain contour lines. Contour lines for a func3on with two variables are lines in which the func3on has a constant value

5 2D Poten>al Energy Surface (PES) Another example You will do a few exercises with contour plots in a worksheet aier lecture today.

6 Poten>al Energy Surface: a diatomic using Internal Coordinates Poten>al Energy Bond length What is the most probable configura>on of the diatomic shown above? Geometry Op5miza5on Finding a cri5cal point or local minima on the PES

7 Poten>al Energy Surface: a diatomic using Internal Coordinates Poten>al Energy F(r) Bond length What does the Force mean when using Internal Coordinates? If the bond is compressed the force will make the bond length longer and the sign is posi3ve

8 Poten>al Energy Surface: a diatomic using Internal Coordinates Poten>al Energy F(r) Bond length What does the Force mean when using Internal Coordinates? If the bond is stretched the force will make the bond length shorter and the sign is nega3ve

9 Cartesian Coordinates Posi>ons of your molecular systems 2 x,y,z x 2,y 2,z 2 Forces on each atom in your system F F = F x, F y, F z 2 F2 F 2 = F x2, F y2, F z2

10 2 Poten>al Energy Surface: a diatomic using Cartesian Coordinates Poten>al Energy F(r) 2 Bond length What does the Force mean when using Internal Coordinates? If the bond is compressed the force will make the bond length longer

11 Poten>al Energy Surface: a diatomic using Internal Coordinates Poten>al Energy 2 F(r) Bond length What does the Force mean when using Internal Coordinates? If the bond is stretched the force will make the bond length shorter and the sign is nega3ve

12 Force Given a D poten>al energy surface, V(r) The force is, F(r) = d dr V(r)

13 Where x F(r) = Force Given a 2D poten>al energy surface, V(x, y) The force is, x V(x, y), y V(x, y) is the par>al deriva>ve with respect to the variable x A par>al deriva>ve is the deriva>ve with respect to only one variable when your func>on is high dimensional.

14 F(r) = Where x Force Given a 2D poten>al energy surface, V(x, y) The force is, x V(x, y), y Gradient V(x, y) = V(x, y) is the par>al deriva>ve with respect to the variable x A par>al deriva>ve is the deriva>ve with respect to only one variable when your func>on is high dimensional.

15 Gradient A gradient is a deriva>ve for a func>on which has more than one variable. The gradient will be vector with dimensions that are equal to the number of input variables in your func>on. Given the func>on: V(x, y) the gradient of V(x,y) is, V(x, y) = x V, y V Where x is the par>al deriva>ve with respect to the variable x

16 Gradient A gradient is a deriva>ve for a func>on which has more than one variable. The gradient will be vector with dimensions that are equal to the number of input variables in your func>on. Given the func>on: V(x, y, z) the gradient of V(x,y) is, V(x, y, z) = x V, y V, z V Where x is the par>al deriva>ve with respect to the variable x

17 Poten>al Energy Surface: a diatomic using Cartesian Coordinates Poten>al Energy V(x, y, z, x 2, y 2, z 2 ) F(r) Posi>ons of your molecular systems 2 x,y,z x 2,y 2,z 2 2 Bond length F = x V, y V, z V, x 2 V, y 2 V, z 2 V

18 Review: Vectors Vector: a quan>ty that contains both a magnitude and a direc>on. Given v = x, x 2, x 3,...x N The magnitude of the en5re vector, v, is v = x 2 + x 2 + x x N The normalized vector, v, is ˆv = v v

19 Normalized vectors The normalized vector is ˆv = The normalized vector is also known as the unit vector The magnitude of a unit vector is always one but is in the same direc>on of vector, v. If you want to change the magnitude of vector v, or rescale the magnitude to M us the following: vm = Mˆv v v

20 D Algorithm for Gradient Descent The general procedure for numerical geometry op>miza>on is as follows:. Calculate the force on all atoms for some configura>on of an atomic system. 2. If the magnitude of the force is less than threshold, you have found a cri>cal point STOP. 3. If not, move the atoms such that they go towards a cri>cal points 4. Repeat. r n+ = r n +αf(r n )

21 x, y n+ = x, y n +α F x, F y n 2D Algorithm for Gradient Descent The general procedure for numerical geometry op>miza>on is as follows:. Calculate the force on all atoms for some configura>on of an atomic system. 2. If the magnitude of the force is less than threshold, you have found a cri>cal point STOP. 3. If not, move the atoms such that they go towards a cri>cal points 4. Repeat.

22 High Dimensional Algorithm for Gradient Descent The general procedure for numerical geometry op>miza>on is as follows:. Calculate the force on all atoms for some configura>on of an atomic system. 2. If the magnitude of the force is less than threshold, you have found a cri>cal point STOP. 3. If not, move the atoms such that they go towards a cri>cal points # # # # # # "# x y z x 2 y 2 z 2 : : x A x A x A 4. Repeat. $ & & & & & & % & n+ # # # = # # # "# x y z x 2 y 2 z 2 : : x A x A x A $ & # & # & # & +α# & # & # %& # n " F x F y F z F x2 F y2 F z2 : : F xa F ya F za Where A is the number of atoms in the chemical system $ & & & & & & & % n

23 Poten>al Energy Surface: a diatomic using Cartesian Coordinates Poten>al Energy V(x, y, z, x 2, y 2, z 2 ) F(r) Posi>ons of your molecular systems 2 x,y,z x 2,y 2,z 2 2 F = x V, Bond length y V, z V, ENTIRE FORCE VECTOR x 2 V, y 2 V, z 2 V

24 Vector Review from Tuesday 2/20

25 Magnitude of Vector y v v =, v = = 2 Use Pythagorean theorem to calculate the length of the third side of triangle, or magnitude of vector, v.

26 Direc>on of Vector: 2 y Normalized vector v =, v = = 2 v v = 2, ˆv = 2 ˆv = 2, 2 We now have a vector in the same direc3on, but with a magnitude of

27 Direc>on of Vector: Rescale length of vector y v =, ˆv = v M = Mˆv 2, v 0.2 = , 2 v 0.2 = 0.2 2, v 0.2 = 0.4, 0.4

28 Vector Problem: How to interpret Force vector F =, F 2 =, F 2 F F 2 - -

29 Vector Problem Force Vectors You can translate vectors Next we will translate the F vector to act on atom - F F 2 -

30 Vector Problem Force Vectors Atomic posi>ons F 2 - F You can translate vectors Next we will translate the F and F 2 vectors to act on atom and 2 No>ce that the scales are different on these two examples.