# Solution¤

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```
diffrax.Solution (AbstractPath)
```

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The solution to a differential equation.

**Attributes:**

`t0`

: The start of the interval that the differential equation was solved over.`t1`

: The end of the interval that the differential equation was solved over.`ts`

: Some ordered collection of times. Might be`None`

if no values were saved. (i.e. just`diffeqsolve(..., saveat=SaveAt(dense=True))`

is used.)`ys`

: The value of the solution at each of the times in`ts`

. Might`None`

if no values were saved.`stats`

: Statistics for the solve (number of steps etc.).`result`

: Integer specifying the success or cause of failure of the solve. A value of`0`

corresponds to a successful solve. Any other value is a failure. A human-readable message can be obtained by looking up messages via`diffrax.RESULTS[<integer>]`

.`solver_state`

: If saved, the final internal state of the numerical solver.`controller_state`

: If saved, the final internal state for the step size controller.`made_jump`

: If saved, the final internal state for the jump tracker.

Note

If `diffeqsolve(..., saveat=SaveAt(steps=True))`

is set, then the `ts`

and `ys`

in the solution object will be padded with `NaN`

s, out to the value of
`max_steps`

passed to `diffrax.diffeqsolve`

.

This is because JAX demands that shapes be known statically ahead-of-time. As we do not know how many steps we will take until the solve is performed, we must allocate enough space for the maximum possible number of steps.

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`t0: Union[float, int]`

`dataclass-field`

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`t1: Union[float, int]`

`dataclass-field`

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`ts: Optional[PyTree[Real[Array, '?times'], "S"]]`

`dataclass-field`

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`ys: Optional[PyTree[Shaped[Array, '?times ?*shape'], "S ..."]]`

`dataclass-field`

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`stats: dict[str, Any]`

`dataclass-field`

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`result: RESULTS`

`dataclass-field`

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`solver_state: Optional[PyTree]`

`dataclass-field`

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`controller_state: Optional[PyTree]`

`dataclass-field`

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`made_jump: Optional[bool]`

`dataclass-field`

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`evaluate(self, t0: Union[float, int], t1: Union[float, int] = None, left: bool = True) -> PyTree[Shaped[Array, '?*shape'], "Y"]`

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If dense output was saved, then evaluate the solution at any point in the
region of integration `self.t0`

to `self.t1`

.

**Arguments:**

`t0`

: The point to evaluate the solution at.`t1`

: If passed, then the increment from`t0`

to`t1`

is returned. (`=evaluate(t1) - evaluate(t0)`

)`left`

: When evaluating at a jump in the solution, whether to return the left-limit or the right-limit at that point.

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`derivative(self, t: Union[float, int], left: bool = True) -> PyTree[Shaped[Array, '?*shape'], "Y"]`

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If dense output was saved, then calculate an **approximation** to the
derivative of the solution at any point in the region of integration `self.t0`

to `self.t1`

.

That is, letting \(y\) denote the solution over the interval `[t0, t1]`

, then
this calculates an approximation to \(\frac{\mathrm{d}y}{\mathrm{d}t}\).

(This is *not* backpropagating through the differential equation -- that
typically corresponds to e.g. \(\frac{\mathrm{d}y(t_1)}{\mathrm{d}y(t_0)}\).)

Example

For an ODE satisfying

\(\frac{\mathrm{d}y}{\mathrm{d}t} = f(t, y(t))\)

then this value is approximately equal to \(f(t, y(t))\).

Warning

This value is generally not very accurate. Differential equation solvers are usually designed to produce splines whose value is close to the true solution; not to produce splines whose derivative is close to the derivative of the true solution.

If you need accurate derivatives for the solution of an ODE, it is usually
best to calculate `vector_field(t, sol.evaluate(t), args)`

. That is, to
pay the extra computational cost of another vector field evaluation, in
order to get a more accurate value.

Put precisely: this `derivative`

method returns the *derivative of the
numerical solution*, and *not* an approximation to the derivative of the
true solution.

**Arguments:**

`t`

: The point to calculate the derivative of the solution at.`left`

: When evaluating at a jump in the solution, whether to return the left-limit or the right-limit at that point.