002.md

# [ADR-002] Inserting dynamics

- **Status**: proposed
- **Deciders**: @redeboer @spflueger

```{autolink-skip} file

```

## Context and problem statement

Physics models usually include assumptions that simplify the structure of the model. For
example, splitting a model into a product of independent parts, in which every part
contains a certain responsibility. In case of partial wave amplitude models, we can make
a separation into a spin part and a dynamical part. While the spin part controls the
probability w.r.t angular kinematic variables, the dynamics controls the probability on
variable like the invariant mass of states.

Generally, a dynamics part is simply a function, which is defined in complex space, and
consists of:

- a mathematical **expression** (`sympy.Expr`)
- a set of **parameters** in that expression that can be tweaked (optimized)
- a set of (kinematic) **variables** to which the expression applies

### Technical story

- [ComPWA/expertsystem#124](https://github.com/ComPWA/expertsystem/issues/124): see {ref}`adr/002:Issues with existing set-up`.
- [ComPWA/expertsystem#440](https://github.com/ComPWA/expertsystem/issues/440): no way to supply custom dynamics. Or at least,
  {mod}`tensorwaves` does not understand those custom dynamics.
- [ADR-001](./001.md): parameters _and_ variables are to be expressed as
  `sympy.Symbol`s.
- [ComPWA/expertsystem#454](https://github.com/ComPWA/expertsystem/issues/454): dynamics are specified as a mapping of `sympy.Function`
  to a `sympy.Expr`, but now there is no way to supply those `sympy.Expr`s with expected
  `sympy.Symbol`s (parameters and variables).

### Issues with existing set-up

- There is no clear way to apply dynamics functions to a specific decaying particle,
  that is, to a specific edge of the `StateTransitionGraph`s (`STG`). Currently, we just
  work with a mapping of `Particle`s to some dynamics expression, but this is not
  feasible when there there are identical particles on the edges.
- The set of variables to which a dynamics expression applies, is determined by the
  position within the `STG` that it is applied to. For instance, a relativistic
  Breit-Wigner that is applied to the resonance in some 1-to-3 isobar decay (described
  by an `STG` with final state edges 2, 3, 4 and intermediate edge 1) would work on the
  invariant mass of edge 3 and 4 (`mSq_3+4`).
- Just like variables, parameters need to be identifiable from their position within the
  `STG` (take a relativistic Breit-Wigner _with form factors_, which would require
  break-up momentum as a parameter), but also require some suggested starting value
  (e.g. expected pole position). These starting values are usually taken from the edge
  and node properties within the `STG`.

## Decision drivers

The following points are nice to have or can influence the decision but are not
essential and can be part of the users responsibility.

1. The parameters that a dynamics functions requires, are registered automatically and
   linked together.
2. Kinematic variables used in dynamics functions are also linked appropriately.
3. It is easy to define custom dynamics (no boilerplate code).

### Solution requirements

1. It is easy to apply dynamics to specific components of the `STG`s. Note: it's
   important that the dynamics can be applied to resonances of some **selected** graphs
   and not generally all graphs in which the resonance appears.
2. Where possible, suggested (initial) parameter values are provided as well.
3. It is possible to use and inspect the dynamics expression itself independently from
   the `expertsystem`.
4. Follow open-closed principle. Probably the most important decision driver. The
   solution should be flexible enough to handle any possible scenario, without having to
   change the interface defined in requirement 1!

## Considered solutions

### Group 1: expression builder

To satisfy [requirement 1](#solution-requirements), we propose the following syntax:

```python
# model: ModelInfo
# graph: StateTransitionGraph
model.set_dynamics(graph, edge_id=1, expression_builder)
```

:::{toggle}

Another style would be to have `ModelInfo` contain a reference to the list of
`StateTransitionGraph`s. The user then needs some other way to express which edges to
apply the dynamics function to:

```python
model.set_dynamics(
    filter=lambda p: p.name.startswith("f(0)"),
    edge_id=1,
    expression_builder,
)
```

:::

Here, `expression_builder` is some function or method that can create a dynamics
expression. It can also be a class that contains both the implementation of the
expression and a static method to build itself from a `StateTransitionGraph`.

The dynamics expression needs to be formulated in such a way that it satisfies
[the rest of the requirements](#solution-requirements). The following options illustrate
three different ways of formulating a dynamics expression, each taking a relativistic
Breit-Wigner and a relativistic Breit-Wigner _with form factor_ as example.

```{toctree}
---
maxdepth: 1
---
002/composition
002/function
002/expr
```

### Group 2: expression-only

A second branch of solutions would propose the following interface:

```python
# model: ModelInfo
# graph: StateTransitionGraph
model.set_dynamics(graph, edge_id=1, expression)
```

The key difference is the usage of general sympy expression `sympy.Expr` as an argument
instead of constructing this through some builder object.

## Solution evaluation

### 1: Expression builder

All of the solutions have the drawback arising from the choice of interface using a
`expression_builder`. This enforces the logic of correctly coupling variables and
parameters into these builders. This is extremely hard to get right, since the code has
to be able to handle arbitrarily complex models. And always knowing what the user would
like to do is more or less impossible. Therefore it is much better to use a already
built expression that is assumed to be correctly built (see
[solution group 2](#group-2-expression-only)).

All of the solutions in this group also have the following additional drawbacks. These
are however more related to the correct building of the dynamics expression:

- There is an implicit assumption on the signature of the expression: the first
  arguments are assumed to be the (kinematic) `variables` and the remaining arguments
  are `parameters`. In addition, the arguments cannot be keywords, but have to be
  positional.
- The number of `variables` and `parameters` is only verified at runtime (no static
  typing, other than a check that each of the elements is `sympy.Symbol`).

Composition is the cleanest design, but is less in tune with the design of {mod}`sympy`.
{doc}`002/function` and {doc}`002/expr` follow {mod}`sympy` implementations, but result
in an obscure inheritance hierarchy with implicit conventions. This can result in some
nasty bugs, for instance if one were to `__call__` method in either the `sympy.Function`
or `sympy.Expr` implementation.

Pros and Cons that are specific to each of the implementations are listed below.

#### {doc}`002/composition`

- **Positive**
  - Implementation of the expression is transparent
- **Negative**
  - {ref}`adr/002/composition:Alternative signature`.
  - The only way to see that `relativistic_breit_wigner_from_graph` is the builder for
    `relativistic_breit_wigner`, is from its name. This makes it implementing custom
    dynamics inconvenient and error-prone.
  - Signature of the builder can only be checked with a {class}`~typing.Protocol`, see
    {ref}`adr/002/composition:Type checking`.

#### {doc}`002/function`

- **Positive**
  - `DynamicsFunction` behaves as a {class}`~sympy.core.function.Function`
  - Implementation of the builder is kept together with the implementation of the
    expression.
- **Negative**
  - It's not possible to identify variables and parameters

#### {doc}`002/expr`

- **Positive**
  - When
    {doc}`recursing through the amplitude model <sympy:tutorials/intro-tutorial/manipulation>`,
    it is still possible to identify instances of `DynamicsExpr` (before `doit()` has
    been called).
  - Additional properties and methods can be added and carried around by the class.
- **Negative**
  - Boilerplate code required when implementing custom dynamics
  - {ref}`adr/002/expr:Issue with lambdify`

### 2: Expression-only

**Positive**: This choice of interface follows the principle of SOLID more than solution
group 1. By handing a complete expression of the dynamics to the setter, its sole
responsibility is to insert this expression at the correct place in the full model
expression.

**Negative**: There are no direct negative aspects to this solution, as it just splits
up responsibilities. The construction of the expression with the correct linking of
parameters and initial values etc has to be performed by some other code. This code is
subject to the same issues mentioned in the individual solutions of group 1.

## Decision outcome

Use a composition based solution from group 2.

Important is the definition of the interface following solution group 2. This ensures to
be open-closed and keep the responsibilities separated.

The `expertsystem` favors **composition over inheritance**: we intend to use inheritance
only to define interfaces, not to insert behavior. As such, the design of the
`expertsystem` is fundamentally different than that of SymPy. That's another reason to
favor composition here: the interfaces are not determined by the dependency and instead
remain {doc}`contained within the dynamics class <002/composition>`.

We decide to keep responsibilities as separated as possible. This means that:

1. The only responsibility of `set_dynamics` method is to attribute some expression
   (`sympy.Expr`) the correct symbol within the complete amplitude model. For now, this
   position is specified using some `StateTransitionGraph` and an edge ID, but this
   syntax may be improved later (see [ComPWA/expertsystem#458](https://github.com/ComPWA/expertsystem/issues/458)ps://github.com/ComPWA/expertsystem/issues/458)):

   ```python
   def set_dynamics(
       self,
       graph: StateTransitionGraph,
       edge_id: int,
       expression: sp.Expr,
   ) -> None:
       # dynamics_symbol = graph + edge_id
       # self.dynamics[dynamics_symbol] = expression
       pass
   ```

   It is assumed that the `expression` is correct.

2. The user has the responsibility of formulating the `expression` with the correct
   symbols. To aid the user in the construction of such expressions some building code
   can handle some of the common tasks, such as

   - A `VariablePool` can facilitate finding the correct symbol names (to avoid typos).

     ```python
     mass = variable_pool.get_invariant_mass(graph, edge_id)
     ```

   - A `dynamics` module provides descriptions of common line-shapes as well as some
     helper functions. An example would be:

     ```python
     inv_mass, mass0, gamma0 = build_relativistic_breit_wigner(graph, edge_id, particle)
     rel_bw: sympy.Expr = relativistic_breit_wigner(inv_mass, mass0, gamma0)
     model.set_dynamics(graph, edge, rel_bw)
     ```

3. The `SympyModel` has the responsibility of defining a the full model in terms of an
   expression and keeping track of variables and parameters, for instance:

   ```python
   from __future__ import annotations
   from attrs import define, field
   import sympy as sp


   @define
   class SympyModel:
       top: sp.Expr
       # intensities: dict[sp.Symbol, sp.Expr] = field(factory=dict)
       # amplitudes: dict[sp.Symbol, sp.Expr] = field(factory=dict)
       dynamics: dict[sp.Symbol, sp.Expr] = field(factory=dict)
       parameters: set[sp.Symbol] = field(factory=set)
       variables: set[sp.Symbol] = field(factory=set)  # or: VariablePool

       def full_expression(self) -> sp.Expr: ...
   ```