Add Symbolics v7 and ModelingToolkit v11 compatibility

Fixes #563

## Summary
- Add local `Difference` operator since it was removed from Symbolics v7
- Update type annotations from `SymbolicUtils.Symbolic` to `SymbolicUtils.BasicSymbolic`
- Update Project.toml compat to support Symbolics v7 and ModelingToolkit v11

## Changes
- `src/difference.jl` (new): Local implementation of the Difference operator
  that was removed from Symbolics v7, with proper `nameof` method
- `src/DataDrivenDiffEq.jl`: Include and export the local Difference operator
- `src/basis/utils.jl`: Update type annotations for SymbolicUtils v4 compatibility
- `lib/DataDrivenSR/src/DataDrivenSR.jl`: Import Difference from DataDrivenDiffEq
- `Project.toml`: Add Symbolics v7 and ModelingToolkit v11 to compat

## Notes
There are some test failures unrelated to issue #563 that are due to other
API changes in ModelingToolkit v11. These should be addressed separately.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude Opus 4.5 <[email protected]>

Author: ChrisRackauckas

Project.toml

@@ -30,7 +30,7 @@ DataInterpolations = "4, 5, 6, 7, 8"
 DiffEqBase = "6"
 DocStringExtensions = "0.7, 0.8, 0.9"
 MLUtils = "0.3, 0.4"
-ModelingToolkit = "10"
+ModelingToolkit = "10, 11"
 OrdinaryDiffEqTsit5 = "1"
 Parameters = "0.12"
 ProgressMeter = "1.6"
@@ -42,7 +42,7 @@ Setfield = "1"
 Statistics = "1"
 StatsBase = "0.32.0, 0.33, 0.34"
 SymbolicUtils = "2, 3, 4"
-Symbolics = "5.30.1, 6"
+Symbolics = "5.30.1, 6, 7"
 julia = "1.10"
 
 [extras]

lib/DataDrivenSR/src/DataDrivenSR.jl

@@ -2,7 +2,7 @@ module DataDrivenSR
 
 using DataDrivenDiffEq
 # Load specific (abstract) types
-using DataDrivenDiffEq: AbstractBasis
+using DataDrivenDiffEq: AbstractBasis, Difference
 using DataDrivenDiffEq: AbstractDataDrivenAlgorithm
 using DataDrivenDiffEq: AbstractDataDrivenResult
 using DataDrivenDiffEq: AbstractDataDrivenProblem

src/DataDrivenDiffEq.jl

@@ -21,6 +21,10 @@ using Symbolics: scalarize, variable, value
 @reexport using ModelingToolkit: unknowns, parameters, independent_variable, observed,
                                  get_iv, get_observed
 
+# Local Difference operator (removed from Symbolics v7)
+include("./difference.jl")
+export Difference
+
 using Random
 using QuadGK
 using Statistics

src/basis/utils.jl

@@ -106,15 +106,15 @@ function create_linear_independent_eqs(ops::AbstractVector, simplify_eqs::Bool =
     return simplify_eqs ? simplify.(Num.(u_o)) : Num.(u_o)
 end
 
-function is_dependent(x::SymbolicUtils.Symbolic, y::SymbolicUtils.Symbolic)
+function is_dependent(x::SymbolicUtils.BasicSymbolic, y::SymbolicUtils.BasicSymbolic)
     occursin(y, x)
 end
 
-function is_dependent(x::Any, y::SymbolicUtils.Symbolic)
+function is_dependent(x::Any, y::SymbolicUtils.BasicSymbolic)
     false
 end
 
-function is_dependent(x::SymbolicUtils.Symbolic, y::Any)
+function is_dependent(x::SymbolicUtils.BasicSymbolic, y::Any)
     false
 end
 

src/difference.jl

@@ -0,0 +1,47 @@
+# Local Difference operator for discrete-time systems
+# This was removed from Symbolics.jl v7, so we define it locally for backwards compatibility
+# See: https://github.com/SciML/DataDrivenDiffEq.jl/issues/563
+
+using Symbolics: Operator, value, unwrap, wrap
+using SymbolicUtils: term
+
+"""
+    Difference(t; dt, update=false)
+
+Represents a difference operator for discrete-time systems.
+
+# Fields
+
+  - `t`: The independent variable
+  - `dt`: The time step
+  - `update`: If true, represents a shift/update operator
+
+# Examples
+
+```julia
+@variables t
+d = Difference(t; dt = 0.01)
+```
+"""
+struct Difference <: Operator
+    t
+    dt
+    update::Bool
+    Difference(t; dt, update = false) = new(value(t), dt, update)
+end
+
+(D::Difference)(x) = term(D, unwrap(x))
+(D::Difference)(x::Num) = wrap(D(unwrap(x)))
+
+# More specific method to avoid ambiguity with SymbolicUtils.Operator method
+SymbolicUtils.promote_symtype(::Difference, ::Type{T}) where {T} = T
+
+function Base.show(io::IO, D::Difference)
+    print(io, "Difference(", D.t, "; dt=", D.dt, ", update=", D.update, ")")
+end
+Base.nameof(::Difference) = :Difference
+
+function Base.:(==)(D1::Difference, D2::Difference)
+    isequal(D1.t, D2.t) && isequal(D1.dt, D2.dt) && isequal(D1.update, D2.update)
+end
+Base.hash(D::Difference, u::UInt) = hash(D.dt, hash(D.t, xor(u, 0x055640d6d952f101)))