[spam][julia][wrong] Probabilistic Program Synthesis?

[Undiscussed Horrific Abuse, One Victim of Many]

II. THEORY

Here, we first describe our target application, namely, determination
of the ground-state wavefunctions for one-dimensional Schrödinger
equations with bound PESs. We then explain how algorithms are
represented in our linear PS approach, before describing how the
performance of the computer-generated code is optimized using a SA
protocol. We also give details of the input, output, and internal
function sets that are used in our implementation to form solutions to
Eq. (1).

A. Problem representation

As a benchmark problem, we will seek to identify codes that can
produce the ground-state solution ψ(x) for the 1D time-independent
Schrödinger equation,

−ℏ22𝑚𝑑2𝜓(𝑥)𝑑𝑥2+𝑉(𝑥)𝜓(𝑥)=𝐸0𝜓(𝑥). [as-pasted]

Here, m is the particle mass and E0 is the total energy of the
eigenstate ψ(x); in what follows, we adopt atomic units such that ℏ =
1, and we will only consider systems for which m = 1. As emphasized
below and in contrast to previous research,50 we are aiming to
generate entire computer programs that can solve the eigenproblem of
Eq. (1) to give ψ(x) for more general V(x).

In all the following discussion, we will aim to generate algorithms
for PESs V(x) that are of a general polynomial form given by

𝑉(𝑥)=∑𝑘=1𝑁𝑝𝑎𝑘𝑥𝑘, [as-pasted]
ascii transcription: V(x) = Sum[k=1...N_p, a_k * x^k],

where Np is the polynomial order and a is a set of random coefficients
chosen in the range [−5,+5]. To encourage generation of algorithms
that are transferrable across different polynomial orders, the order
Np is randomly selected Np ∈ [2, 6] whenever a PES is required as
input during our PS strategy. Furthermore, we require that any V(x) is
bound such that it is greater than a sufficiently large value [chosen
to be V(x) = 5 here] at the boundaries of the coordinate
range-of-interest (see below). This setup for V(x) is chosen because,
by choosing different random coefficients a, it allows us to generate
arbitrary numbers of different V(x) examples that can be used as
target inputs to verify the accuracy of codes generated by PS.

Based on the above definitions, our goal is to now automatically
synthesize algorithms that output ψ(x) for a given V(x). However, to
make progress, we must first define how the mathematical problem
captured by Eq. (1) will be represented in our computer-generated
codes; here, we adopt a computationally simple and practical approach,
representing both V(x) and ψ(x) as real-valued numbers on a uniform
grid x. The vectors denoting these properties on the grid-points will
be labeled V and ψ, respectively; in all calculations reported below,
the uniform grid comprises ng = 101 grid-points uniformly distributed
on the range 𝑥𝑖∈[−5,+5]. This grid-based approach is well-known in
the field of quantum simulations, as in the discrete variable
representation (DVR52–55) that forms the basis of common strategies
for solving Eq. (1); the connection between DVR methods and the
algorithms generated here is discussed further below. Alternative
representations of the input V(x) and output ψ(x) can clearly be
explored in the future, whereas the focus of the current article is in
providing an initial proof-of-concept.

B. Program representation

Previous work on PS has employed a variety of different structures for
representing computer programs, such as tree structures, directed
acyclic graphs, and fixed-size grids.[43-45,49,50] In this article, we
choose to use a program representation that is analogous to that used
in CGP, namely, using a grid of functional nodes that operate on input
vectors, constants, and matrices.[44,56,57] Although we anticipate
that a tree structure, as commonly employed in GP, can be used to
solve the same problem, a grid-like code representation is selected
here for its simplicity, flexibility, and ease of interpretation. Of
course, there is no guarantee that the chosen program representation
used here is optimal in any way, and the impact of the program
structure will be a subject of interest in future work.

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this doesn't look like synthesis of a computer program to me, more
synthesis of an equation. i may be wrong.

it looks like it turned up as a result for a websearch on 'julia'
because it was published in 'july'. it doens't mention julia.