Toy Examples with Code

using SymbolicRegression
using MLJ

1. Simple search

Here's a simple example where we find the expression 2 cos(x4) + x1^2 - 2.

X = 2randn(1000, 5)
y = @. 2*cos(X[:, 4]) + X[:, 1]^2 - 2

model = SRRegressor(
    binary_operators=[+, -, *, /],
    unary_operators=[cos],
    niterations=30
)
mach = machine(model, X, y)
fit!(mach)

Let's look at the returned table:

r = report(mach)
r

We can get the selected best tradeoff expression with:

r.equations[r.best_idx]

2. Custom operator

Here, we define a custom operator and use it to find an expression:

X = 2randn(1000, 5)
y = @. 1/X[:, 1]

my_inv(x) = 1/x

model = SRRegressor(
    binary_operators=[+, *],
    unary_operators=[my_inv],
)
mach = machine(model, X, y)
fit!(mach)
r = report(mach)
println(r.equations[r.best_idx])

3. Multiple outputs

Here, we do the same thing, but with multiple expressions at once, each requiring a different feature. This means that we need to use MultitargetSRRegressor instead of SRRegressor:

X = 2rand(1000, 5) .+ 0.1
y = @. 1/X[:, 1:3]

my_inv(x) = 1/x

model = MultitargetSRRegressor(; binary_operators=[+, *], unary_operators=[my_inv])
mach = machine(model, X, y)
fit!(mach)

The report gives us lists of expressions instead:

r = report(mach)
for i in 1:3
    println("y[$(i)] = ", r.equations[i][r.best_idx[i]])
end

4. Plotting an expression

For now, let's consider the expressions for output 1 from the previous example: We can get a SymbolicUtils version with:

using SymbolicUtils

eqn = node_to_symbolic(r.equations[1][r.best_idx[1]])

We can get the LaTeX version with Latexify:

using Latexify

latexify(string(eqn))

We can also plot the prediction against the truth:

using Plots

ypred = predict(mach, X)
scatter(y[1, :], ypred[1, :], xlabel="Truth", ylabel="Prediction")

5. Other types

SymbolicRegression.jl can handle most numeric types you wish to use. For example, passing a Float32 array will result in the search using 32-bit precision everywhere in the codebase:

X = 2randn(Float32, 1000, 5)
y = @. 2*cos(X[:, 4]) + X[:, 1]^2 - 2

model = SRRegressor(binary_operators=[+, -, *, /], unary_operators=[cos], niterations=30)
mach = machine(model, X, y)
fit!(mach)

we can see that the output types are Float32:

r = report(mach)
best = r.equations[r.best_idx]
println(typeof(best))
# Expression{Float32,Node{Float32},...}

We can also use Complex numbers (ignore the warning from MLJ):

cos_re(x::Complex{T}) where {T} = cos(abs(x)) + 0im

X = 15 .* rand(ComplexF64, 1000, 5) .- 7.5
y = @. 2*cos_re((2+1im) * X[:, 4]) + 0.1 * X[:, 1]^2 - 2

model = SRRegressor(
    binary_operators=[+, -, *, /],
    unary_operators=[cos_re],
    maxsize=30,
    niterations=100
)
mach = machine(model, X, y)
fit!(mach)

6. Dimensional constraints

One other feature we can exploit is dimensional analysis. Say that we know the physical units of each feature and output, and we want to find an expression that is dimensionally consistent.

We can do this as follows, using DynamicQuantities to assign units. First, let's make some data on Newton's law of gravitation:

using DynamicQuantities
using SymbolicRegression

M = (rand(100) .+ 0.1) .* Constants.M_sun
m = 100 .* (rand(100) .+ 0.1) .* u"kg"
r = (rand(100) .+ 0.1) .* Constants.R_earth

G = Constants.G

F = @. (G * M * m / r^2)

(Note that the u macro from DynamicQuantities will automatically convert to SI units. To avoid this, use the us macro.)

Now, let's ready the data for MLJ:

X = (; M=M, m=m, r=r)
y = F

Since this data has such a large dynamic range, let's also create a custom loss function that looks at the error in log-space:

function loss_fnc(prediction, target)
    # Useful loss for large dynamic range
    scatter_loss = abs(log((abs(prediction)+1e-20) / (abs(target)+1e-20)))
    sign_loss = 10 * (sign(prediction) - sign(target))^2
    return scatter_loss + sign_loss
end

Now let's define and fit our model:

model = SRRegressor(
    binary_operators=[+, -, *, /],
    unary_operators=[square],
    elementwise_loss=loss_fnc,
    complexity_of_constants=2,
    maxsize=25,
    niterations=100,
    populations=50,
    dimensional_constraint_penalty=10^5,
)
mach = machine(model, X, y)
fit!(mach)

You can observe that all expressions with a loss under our penalty are dimensionally consistent! (The "[?]" indicates free units in a constant, which can cancel out other units in the expression.) For example,

"y[m s⁻² kg] = (M[kg] * 2.6353e-22[?])"

would indicate that the expression is dimensionally consistent, with a constant "2.6353e-22[m s⁻²]".

Note that you can also search for dimensionless units by settings dimensionless_constants_only to true.

7. Working with Expressions

Expressions in SymbolicRegression.jl are represented using the Expression{T,Node{T},...} type, which provides a more robust way to combine structure, operators, and constraints. Here's an example:

using SymbolicRegression

# Define options with operators and structure
options = Options(
    binary_operators=[+, -, *],
    unary_operators=[cos],
    expression_options=(
        structure=TemplateStructure(),
        variable_constraints=Dict(1 => [1, 2], 2 => [2])
    )
)

# Create expression nodes with constraints
operators = options.operators
variable_names = ["x1", "x2"]
x1 = Expression(
    Node{Float64}(feature=1),
    operators=operators,
    variable_names=variable_names,
    structure=options.expression_options.structure
)
x2 = Expression(
    Node{Float64}(feature=2),
    operators=operators,
    variable_names=variable_names,
    structure=options.expression_options.structure
)

# Construct and evaluate expression
expr = x1 * cos(x2 - 3.2)
X = rand(Float64, 2, 100)
output = expr(X)

This Expression type, contains both the structure and the operators used in the expression. These are what are returned by the search. The raw Node type (which is what used to be output directly) is accessible with

get_contents(expr)

8. Parametric Expressions

Parametric expressions allow the algorithm to optimize parameters within the expressions during the search process. This is useful for finding expressions that not only fit the data but also have tunable parameters.

To use this, the data needs to have information on which class each row belongs to –- this class information will be used to select the parameters when evaluating each expression.

For example:

using SymbolicRegression
using MLJ

# Define the dataset
X = NamedTuple{(:x1, :x2)}(ntuple(_ -> randn(Float32, 30), Val(2)))
X = (; X..., classes=rand(1:2, 30))
p1 = [0.0f0, 3.2f0]
p2 = [1.5f0, 0.5f0]

y = [
    2 * cos(X.x1[i] + p1[X.classes[i]]) + X.x2[i]^2 - p2[X.classes[i]] for
    i in eachindex(X.classes)
]

# Define the model with parametric expressions
model = SRRegressor(
    niterations=100,
    binary_operators=[+, *, /, -],
    unary_operators=[cos],
    expression_type=ParametricExpression,
    expression_options=(; max_parameters=2),
    parallelism=:multithreading
)

# Train the model
mach = machine(model, X, y)
fit!(mach)

# View the best expression
report(mach)

The final equations will contain parameters that were optimized during training:

eq = report(mach).equations[end]

typeof(eq)

We can also access the parameters of the expression with:

get_metadata(eq).parameters

This example demonstrates how to set up a symbolic regression model that searches for expressions with parameters, optimizing both the structure and the parameters of the expressions based on the provided class information.

9. Additional features

For the many other features available in SymbolicRegression.jl, check out the API page for Options. You might also find it useful to browse the documentation for the Python frontend PySR, which has additional documentation. In particular, the tuning page is useful for improving search performance.

10. Template Expressions

Template expressions allow you to define structured expressions where different parts can be constrained to use specific variables. In this example, we'll create expressions that output pairs of values.

First, let's set up our basic configuration:

using SymbolicRegression
using Random: rand
using MLJBase: machine, fit!, report

options = Options(
    binary_operators=(+, *, /, -),
    unary_operators=(sin, cos)
)
operators = options.operators
variable_names = ["x1", "x2", "x3"]

Now we'll create base expressions for each variable:

x1, x2, x3 = [
    Expression(
        Node{Float64}(feature=i);
        operators=operators,
        variable_names=variable_names
    )
    for i in 1:3
]

The key part is defining our template structure. This determines how different parts of the expression combine:

structure = TemplateStructure{(:f, :g1, :g2)}(;
    # Define how to combine vectors of evaluated expressions
    combine_vectors=e -> map(
        (f, g1, g2) -> (f + g1, f + g2),
        e.f, e.g1, e.g2
    ),
    # Define how to combine strings for printing
    combine_strings=e -> "( $(e.f) + $(e.g1), $(e.f) + $(e.g2) )",
    # Constrain which variables can be used in each part
    variable_constraints=(; f=[1, 2], g1=[3], g2=[3])
)

Let's generate some example data:

X = rand(100, 3) .* 10
# Create pairs of target expressions
y = [
    (sin(X[i, 1]) + X[i, 3]^2, sin(X[i, 1]) + X[i, 3])
    for i in eachindex(axes(X, 1))
]

Now we can set up and train our model:

model = SRRegressor(;
    binary_operators=(+, *),
    unary_operators=(sin,),
    maxsize=25,
    expression_type=TemplateExpression,
    # Pass options used to instantiate expressions
    expression_options=(; structure),
    # Our `y` is 2-tuple of values
    elementwise_loss=((x1, x2), (y1, y2)) -> (y1 - x1)^2 + (y2 - x2)^2
)

mach = machine(model, X, y)
fit!(mach)

After training, we can examine the best expression:

r = report(mach)
best_expr = r.equations[r.best_idx]

# Access individual parts of the template expression
f_part = get_contents(best_expr).f    # Expression using x1 or x2
g1_part = get_contents(best_expr).g1  # Expression using x3
g2_part = get_contents(best_expr).g2  # Expression using x3

The above code demonstrates how template expressions can be used to:

• Define structured expressions with multiple components
• Constrains which variables can be used in each component
• Create expressions that can output multiple values

You can even output custom structs - see the more detailed Template Expression example!