Its derivative f and adds the pair f f to the

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its derivative f 0 , and adds the pair ( f 0 , f ) to the training set. Unlike integration, differentiation is always possible and extremely fast even for very large expressions. As opposed to the forward approach, this method does not depend on an external symbolic integration system. Backward generation with integration by parts ( IBP ). An issue with the backward approach is that it is very unlikely to generate the integral of simple functions like f ( x ) = x 3 sin( x ) . Its integral, F ( x ) = - x 3 cos( x ) + 3 x 2 sin( x ) + 6 x cos( x ) - 6 sin( x ) , a function with 15 operators, has a very low probability of being generated randomly. Besides, the backward approach tends to generate examples where the integral (the solution) is shorter than the derivative (the problem), while forward generation favors the opposite (see Figure 2 in section E in the Appendix). To address this issue, we 4
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leverage integration by parts: given two randomly generated functions F and G , we compute their respective derivatives f and g . If fG already belongs to the training set, we know its integral, and we can compute the integral of Fg as: Z Fg = FG - Z fG Similarly, if Fg is in the training set, we can infer the integral of fG . Whenever we discover the integral of a new function, we add it to the training set. If none of fG or Fg are in the training set, we simply generate new functions F and G . With this approach, we can generate the integrals of functions like x 10 sin( x ) without resorting to an external symbolic integration system. Comparing different generation methods. Table 1 in Section 4.1 summarizes the differences between the three generation methods. The FWD method tends to generate short problems with long solutions (that computer algebras can solve). The BWD approach, on the other hand, generates long problems with short solutions. IBP generates datasets comparable to FWD (short problems and long solutions), without an external computer algebra system. A mixture of BWD and IBP generated data should therefore provide a better representation of problem space, without resorting to external tools. Examples of functions / integrals for the three approaches are given in Table 9 of the Appendix. 3.2 F IRST ORDER DIFFERENTIAL EQUATION (ODE 1) We now present a method to generate first order differential equations with their solutions. We start from a bivariate function F ( x, y ) such that the equation F ( x, y ) = c (where c is a constant) can be analytically solved in y . In other words, there exists a bivariate function f that satisfies ( x, c ) , F ( x, f ( x, c ) ) = c . By differentiation with respect to x , we have that x, c : ∂F ( x, f c ( x )) ∂x + f 0 c ( x ) ∂F ( x, f c ( x )) ∂y = 0 where f c = x 7→ f ( x, c ) . As a result, for any constant c , f c is solution of the first order differential equation: ∂F ( x, y ) ∂x + y 0 ∂F ( x, y ) ∂y = 0 (3) With this approach, we can use the method described in Section C of the appendix to generate arbitrary functions F ( x, y ) analytically solvable in y , and create a dataset of differential equations with their solutions.
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