Mathematical Modeling Tutorial

cds.modeling provides a small symbolic algebra: build expressions, differentiate them symbolically, simplify, export to LaTeX, group them into a MathModel, and solve numerically.

1. Build & Evaluate an Expression

from cds.modeling import Variable, Constant

x = Variable("x")
expr = x ** 2 + 3 * x          # x^2 + 3x
print(expr.evaluate({"x": 2})) # 10.0
print(expr.to_str())           # ((x ** 2) + (3 * x))

2. Symbolic Differentiation

from cds.modeling import Sin, Log, Exp

x = Variable("x")
f = Sin(x) * Exp(x)            # sin(x) * e^x
df = f.diff("x")               # symbolic derivative
print(df.to_str())
print(df.evaluate({"x": 0.0})) # 1.0  (analytically e^0*(sin0+cos0))

The chain, product, and quotient rules are all built in — for Sin, Cos, Exp, Log, Sqrt, Pow, and the arithmetic operators.

3. Simplify & Export to LaTeX

from cds.modeling import Variable

x = Variable("x")
print((x + 0).simplify().to_str())    # x
print((x * 1).simplify().to_str())    # x
print((x ** 0).simplify().to_str())   # 1

expr = (Variable("x") ** Constant(2.0)) / Variable("y")
print(expr.to_latex())                 # \frac{x^{2}}{y}

4. Group Equations into a MathModel

from cds.modeling import MathModel, Variable

t, a = Variable("t"), Variable("a")
v0 = Variable("v0")
model = MathModel(
    name="Kinematics",
    description="Constant-acceleration motion",
    parameters={"a": 9.81},
    variables=["t", "v0", "x0"],
)
model.add_equation("velocity", v0 + a * t)
model.add_equation("position", Variable("x0") + v0 * t + 0.5 * a * t**2)

print(model.evaluate({"t": 2.0, "v0": 0.0, "x0": 0.0}))
# {'velocity': 19.62, 'position': 19.62}

# Jacobian column: how every equation depends on t
for label, d in model.jacobian("t").items():
    print(label, d.to_str())

5. Solve & Fit

from cds.modeling import Variable, solve_equation

# Solve x^2 - 2 = 0  =>  x = sqrt(2)
x = Variable("x")
result = solve_equation(x ** 2 - 2, variable="x", x0=1.0)
print(result.x)            # ~1.4142
print(result.converged)    # True

For fitting a model's parameters to data, see cds.modeling.fit_parameters — it minimizes the residual sum of squares via cds.optimization.gradient_descent.

Run the full demo with python examples/modeling_demo.py.