solve_nivp

A Python library for time integration of nonsmooth ODE/DAE systems—models with abrupt changes such as impacts, switching, or inequality constraints. Such models arise in frictional contact mechanics, piecewise and switching behaviour in circuits, sliding-mode control, and discontinuous rules in finance and energy markets. Classical solvers, which assume smoothness, often require regularisation or very small steps due to the inherent stiffness of these models. solve_nivp builds nonsmooth rules directly into the implicit time-stepping scheme, enabling users to encode constraints and advance the state robustly.

Key features

• Projection-based constraint encoding. Users express set-valued or nonsmooth relations as projections onto convex sets (Coulomb friction cone, sign / normal cone, second-order cone, algebraic constraints). Custom projections need only implement project() and an optional tangent_cone().

• Nonlinear solvers for nonsmooth problems. A semismooth Newton method with Armijo line search and a variational-inequality (VI) fixed-point iteration, both with standard tolerances, safeguards, and iteration diagnostics.

• Implicit integrators. Backward Euler, Trapezoidal, θ-method, a composite TR–BE scheme (Bathe-type, second-order), and an embedded BE–TR error estimator.

• Adaptive step-size control with Richardson extrapolation.

• Optional RL add-on. Exposes the time integrator as a Gym-style environment for learning adaptive step-size policies (TD3 / TQC via Stable Baselines 3).

The library is organised around three interchangeable components—projection, nonlinear solver, and integrator—so that swapping algorithms during experimentation is straightforward. Linear-algebra routines operate on dense or sparse arrays in the SciPy ecosystem.

Installation

Recommended developer install:

python3 -m venv .venv && source .venv/bin/activate
pip install -U pip
pip install -e .[test]

Optional extras:

# RL experiments
pip install -e .[rl]

Quickstart

import numpy as np
from solve_nivp import solve_ivp_ns

# simple smooth rhs: y' = -y
rhs = lambda t, y: -y

t_span = (0.0, 1.0)
y0 = np.array([1.0])

# identity projection, VI solver via composite integrator
sol = solve_ivp_ns(
    fun=rhs,
    t_span=t_span,
    y0=y0,
    method='composite',
    projection='identity',
    solver='VI',
)

print(sol[0][:5], sol[1][:5])  # t, y samples

See examples/ for notebooks on friction stick–slip, bouncing ball (contact/impact), SOC constraints, and sliding-mode control.

Running tests

pytest -q

Building the documentation

cd docs
make clean html

Open docs/_build/html/index.html.

RL experiments (optional)

The RL_Adaption/ folder contains optional experiments (TD3/TQC) for learned adaptivity on challenging nonsmooth problems. Large artifacts are ignored by Git and not required for core installation or testing.

Citation

See CITATION.cff. If you use this software, please cite the JOSS paper once available.

License

MIT License (see LICENSE).