Polars interpolate_by fails when null values are at the beginning or end of a column

Just a quick pedantic point, polars isn't failing to interpolate. You're wanting it to extrapolate.

If you want it to take the last trend and apply it then there's not a built in way to do that but you can do it yourself with expressions with forward_fill and shift.

df.with_columns(
    pl.col("b").fill_null(
        ((pl.col("b")-pl.col("b").shift())/(pl.col("a")-pl.col("a").shift()))
        .forward_fill() *
        (pl.col("a")-pl.col("a").shift()) + pl.col("b").shift()
        )
    )

This extrapolation is interchangeable with your first example's interpolate since there's a constant trend but it'd give you a much different answer if the trend was differing over time (see note at the end).

You can turn this into a function for reuse with

def extrapolate_by(col_to_fill:str|pl.Expr, index_col:str|pl.Expr) -> pl.Expr:
    if isinstance(col_to_fill, str):
        col_to_fill=pl.col(col_to_fill)
    if isinstance(index_col, str):
        index_col=pl.col(index_col)
    return col_to_fill.fill_null(
        ((col_to_fill-col_to_fill.shift())/(index_col-index_col.shift()))
        .forward_fill() *
        ((index_col-index_col.shift())) + col_to_fill.shift()
    )

and then you can simple do df.with_columns(extrapolate_by("b","a")) or if you don't mind not having type hints, you can monkey patch it to pl.Expr and its usage will be chainable

pl.Expr.extrapolate_by = extrapolate_by
df.with_columns(pl.col("b").extrapolate_by("a"))

As is, this only works forward, if you want to fill missing values at the beginning, you'd have to inverse the shift direction and fill direction.

Note on difference between extrapolate and interpolate.

df = pl.DataFrame({
    'a': [1, 2, 3, 4, 5],
    'b': [4, 6, None, 20, 22]
})

df.with_columns(
    interpolate=pl.col("b").interpolate_by("a"),
    extrapolate=pl.col("b").extrapolate_by("a")
    )

shape: (5, 4)
┌─────┬──────┬─────────────┬─────────────┐
│ a   ┆ b    ┆ interpolate ┆ extrapolate │
│ --- ┆ ---  ┆ ---         ┆ ---         │
│ i64 ┆ i64  ┆ f64         ┆ f64         │
╞═════╪══════╪═════════════╪═════════════╡
│ 1   ┆ 4    ┆ 4.0         ┆ 4.0         │
│ 2   ┆ 6    ┆ 6.0         ┆ 6.0         │
│ 3   ┆ null ┆ 13.0        ┆ 8.0         │
│ 4   ┆ 20   ┆ 20.0        ┆ 20.0        │
│ 5   ┆ 22   ┆ 22.0        ┆ 22.0        │
└─────┴──────┴─────────────┴─────────────┘

When you interpolate it takes the known values on either side of the missing value to deduce what the missing value should have been. Extrapolating only takes old information to apply it to the future. In this example the trend changes so there's a big difference between extrapolating and interpolating so just be aware that these aren't always interchangeable.