# Paths¤

Many objects that arise when solving differential equations are paths. That is to say, they are some piecewise continuous function $$f : [t_0, t_1] \to \mathbb{R}^d$$.

Diffrax represents this general notion via the diffrax.AbstractPath class. For example, Brownian motion is a subclass of this. So are the interpolations used to drive neural controlled differential equations.

If you need to create your own path (e.g. to drive a CDE) then you can do so by subclassing diffrax.AbstractPath.

diffrax.AbstractPath

####  diffrax.AbstractPath ¤

Abstract base class for all paths.

Every path has a start point t0 and an end point t1. In between these values it is possible to evaluate the path, or (if it is differentiable, e.g. not Brownian motion) calculate its derivative.

Example

class QuadraticPath(AbstractPath):
@property
def t0(self):
return 0

@property
def t1(self):
return 3

def evaluate(self, t0, t1=None, left=True):
del left
if t1 is not None:
return self.evaluate(t1) - self.evaluate(t0)
return t0 ** 2

##### evaluate(self, t0: Scalar, t1: Optional[Scalar] = None, left: bool = True) -> PyTree abstractmethod ¤

Evaluate the path at any point in the interval $$[t_0, t_1]$$.

Arguments:

• t0: Any point in $$[t_0, t_1]$$ to evaluate the path at.
• t1: If passed, then the increment from t1 to t0 is evaluated instead.
• left: Across jump points: whether to treat the path as left-continuous or right-continuous.

FAQ

Note that we use $$t_0$$ and $$t_1$$ to refer to the overall interval, as obtained via instance.t0 and instance.t1. We use t0 and t1 to refer to some subinterval of $$[t_0, t_1]$$. This is an API that is used for consistency with the rest of the package, and just happens to be a little confusing here.

Returns:

If t1 is not passed:

The value of the path at t0.

If t1 is passed:

The increment of the path between t0 and t1.

##### derivative(self, t: Scalar, left: bool = True) -> PyTree¤

Evaluate the derivative of the path. Essentially equivalent to jax.jvp(self.evaluate, (t,), (jnp.ones_like(t),)) (and indeed this is its default implementation if no other is specified).

Arguments:

• t: Any point in $$[t_0, t_1]$$ to evaluate the derivative at.
• left: Whether to obtain the left-derivative or right-derivative at that point.

Returns:

The derivative of the path.