Training a Neural ODE with Optax¶

We can use the parameter estimation functionality to fit a neural ODE to a time series data set.

In [1]:

"""Train a neural ODE with ProbDiffEq and Optax."""

import jax
import jax.numpy as jnp
import matplotlib.pyplot as plt
import optax
from diffeqzoo import backend, ivps

from probdiffeq import ivpsolve, ivpsolvers, stats
from probdiffeq.util.doc_util import notebook

"""Train a neural ODE with ProbDiffEq and Optax.""" import jax import jax.numpy as jnp import matplotlib.pyplot as plt import optax from diffeqzoo import backend, ivps from probdiffeq import ivpsolve, ivpsolvers, stats from probdiffeq.util.doc_util import notebook

In [2]:

plt.rcParams.update(notebook.plot_style())
plt.rcParams.update(notebook.plot_sizes())

plt.rcParams.update(notebook.plot_style()) plt.rcParams.update(notebook.plot_sizes())

In [3]:

if not backend.has_been_selected:
    backend.select("jax")  # ivp examples in jax

# Catch NaN gradients in CI
# Disable to improve speed
jax.config.update("jax_debug_nans", True)

jax.config.update("jax_platform_name", "cpu")

if not backend.has_been_selected: backend.select("jax") # ivp examples in jax # Catch NaN gradients in CI # Disable to improve speed jax.config.update("jax_debug_nans", True) jax.config.update("jax_platform_name", "cpu")

To keep the problem nice and small, assume that the data set is a trigonometric function (which solve differential equations).

In [4]:

grid = jnp.linspace(0, 1, num=100)
data = jnp.sin(5 * jnp.pi * grid)

plt.plot(grid, data, ".-", label="Data")
plt.legend()
plt.show()

grid = jnp.linspace(0, 1, num=100) data = jnp.sin(5 * jnp.pi * grid) plt.plot(grid, data, ".-", label="Data") plt.legend() plt.show()

In [5]:

def build_loss_fn(vf, initial_values, solver, *, standard_deviation=1e-2):
    """Build a loss function from an ODE problem."""

    @jax.jit
    def loss_fn(parameters):
        """Loss function: log-marginal likelihood of the data."""
        tcoeffs = (*initial_values, vf(*initial_values, t=t0, p=parameters))
        ibm, ssm = ivpsolvers.prior_ibm(tcoeffs, ssm_fact="isotropic")
        ts0 = ivpsolvers.correction_ts0(ssm=ssm)
        strategy = ivpsolvers.strategy_smoother(ssm=ssm)
        solver_ts0 = ivpsolvers.solver(strategy, prior=ibm, correction=ts0, ssm=ssm)
        init = solver_ts0.initial_condition()

        sol = ivpsolve.solve_fixed_grid(
            lambda *a, **kw: vf(*a, **kw, p=parameters),
            init,
            grid=grid,
            solver=solver,
            ssm=ssm,
        )

        observation_std = jnp.ones_like(grid) * standard_deviation
        marginal_likelihood = stats.log_marginal_likelihood(
            data[:, None],
            standard_deviation=observation_std,
            posterior=sol.posterior,
            ssm=sol.ssm,
        )
        return -1 * marginal_likelihood

    return loss_fn

def build_loss_fn(vf, initial_values, solver, *, standard_deviation=1e-2): """Build a loss function from an ODE problem.""" @jax.jit def loss_fn(parameters): """Loss function: log-marginal likelihood of the data.""" tcoeffs = (*initial_values, vf(*initial_values, t=t0, p=parameters)) ibm, ssm = ivpsolvers.prior_ibm(tcoeffs, ssm_fact="isotropic") ts0 = ivpsolvers.correction_ts0(ssm=ssm) strategy = ivpsolvers.strategy_smoother(ssm=ssm) solver_ts0 = ivpsolvers.solver(strategy, prior=ibm, correction=ts0, ssm=ssm) init = solver_ts0.initial_condition() sol = ivpsolve.solve_fixed_grid( lambda *a, **kw: vf(*a, **kw, p=parameters), init, grid=grid, solver=solver, ssm=ssm, ) observation_std = jnp.ones_like(grid) * standard_deviation marginal_likelihood = stats.log_marginal_likelihood( data[:, None], standard_deviation=observation_std, posterior=sol.posterior, ssm=sol.ssm, ) return -1 * marginal_likelihood return loss_fn

In [6]:

def build_update_fn(*, optimizer, loss_fn):
    """Build a function for executing a single step in the optimization."""

    @jax.jit
    def update(params, opt_state):
        """Update the optimiser state."""
        _loss, grads = jax.value_and_grad(loss_fn)(params)
        updates, opt_state = optimizer.update(grads, opt_state)
        params = optax.apply_updates(params, updates)
        return params, opt_state

    return update

def build_update_fn(*, optimizer, loss_fn): """Build a function for executing a single step in the optimization.""" @jax.jit def update(params, opt_state): """Update the optimiser state.""" _loss, grads = jax.value_and_grad(loss_fn)(params) updates, opt_state = optimizer.update(grads, opt_state) params = optax.apply_updates(params, updates) return params, opt_state return update

Construct an MLP with tanh activation¶

Let's start with the example given in the implicit layers tutorial. The vector field is provided by DiffEqZoo.

In [7]:

f, u0, (t0, t1), f_args = ivps.neural_ode_mlp(layer_sizes=(2, 20, 1))


@jax.jit
def vf(y, *, t, p):
    """Evaluate the MLP."""
    return f(y, t, *p)


# Make a solver
tcoeffs = (u0, vf(u0, t=t0, p=f_args))
ibm, ssm = ivpsolvers.prior_ibm(tcoeffs, output_scale=1.0, ssm_fact="isotropic")
ts0 = ivpsolvers.correction_ts0(ssm=ssm)
strategy = ivpsolvers.strategy_smoother(ssm=ssm)
solver_ts0 = ivpsolvers.solver(strategy, prior=ibm, correction=ts0, ssm=ssm)
init = solver_ts0.initial_condition()

f, u0, (t0, t1), f_args = ivps.neural_ode_mlp(layer_sizes=(2, 20, 1)) @jax.jit def vf(y, *, t, p): """Evaluate the MLP.""" return f(y, t, *p) # Make a solver tcoeffs = (u0, vf(u0, t=t0, p=f_args)) ibm, ssm = ivpsolvers.prior_ibm(tcoeffs, output_scale=1.0, ssm_fact="isotropic") ts0 = ivpsolvers.correction_ts0(ssm=ssm) strategy = ivpsolvers.strategy_smoother(ssm=ssm) solver_ts0 = ivpsolvers.solver(strategy, prior=ibm, correction=ts0, ssm=ssm) init = solver_ts0.initial_condition()

In [8]:

sol = ivpsolve.solve_fixed_grid(
    lambda *a, **kw: vf(*a, **kw, p=f_args), init, grid=grid, solver=solver_ts0, ssm=ssm
)

plt.plot(sol.t, sol.u[0], ".-", label="Initial estimate")
plt.plot(grid, data, ".-", label="Data")
plt.legend()
plt.show()

sol = ivpsolve.solve_fixed_grid( lambda *a, **kw: vf(*a, **kw, p=f_args), init, grid=grid, solver=solver_ts0, ssm=ssm ) plt.plot(sol.t, sol.u[0], ".-", label="Initial estimate") plt.plot(grid, data, ".-", label="Data") plt.legend() plt.show()

Set up a loss function and an optimiser¶

Like in the other tutorials, we use Optax.

In [9]:

loss_fn = build_loss_fn(vf=vf, initial_values=(u0,), solver=solver_ts0)
optim = optax.adam(learning_rate=2e-2)
update_fn = build_update_fn(optimizer=optim, loss_fn=loss_fn)

loss_fn = build_loss_fn(vf=vf, initial_values=(u0,), solver=solver_ts0) optim = optax.adam(learning_rate=2e-2) update_fn = build_update_fn(optimizer=optim, loss_fn=loss_fn)

In [10]:

p = f_args
state = optim.init(p)
chunk_size = 25
for i in range(chunk_size):
    for _ in range(chunk_size**2):
        p, state = update_fn(p, state)
    print(
        "Negative log-marginal-likelihood after "
        f"{(i + 1) * chunk_size**2}/{chunk_size**3} steps:",
        loss_fn(p),
    )

p = f_args state = optim.init(p) chunk_size = 25 for i in range(chunk_size): for _ in range(chunk_size**2): p, state = update_fn(p, state) print( "Negative log-marginal-likelihood after " f"{(i + 1) * chunk_size**2}/{chunk_size**3} steps:", loss_fn(p), )

Negative log-marginal-likelihood after 625/15625 steps: 2183.3174

Negative log-marginal-likelihood after 1250/15625 steps: 2139.2583

Negative log-marginal-likelihood after 1875/15625 steps: 2119.1238

Negative log-marginal-likelihood after 2500/15625 steps: 2090.3723

Negative log-marginal-likelihood after 3125/15625 steps: 2055.1594

Negative log-marginal-likelihood after 3750/15625 steps: 2055.1846

Negative log-marginal-likelihood after 4375/15625 steps: 2044.1001

Negative log-marginal-likelihood after 5000/15625 steps: 1969.2017

Negative log-marginal-likelihood after 5625/15625 steps: 1911.1566

Negative log-marginal-likelihood after 6250/15625 steps: 1787.1368

Negative log-marginal-likelihood after 6875/15625 steps: 1582.8367

Negative log-marginal-likelihood after 7500/15625 steps: 1284.9406

Negative log-marginal-likelihood after 8125/15625 steps: 1040.6012

Negative log-marginal-likelihood after 8750/15625 steps: 935.1293

Negative log-marginal-likelihood after 9375/15625 steps: 784.8787

Negative log-marginal-likelihood after 10000/15625 steps: 614.761

Negative log-marginal-likelihood after 10625/15625 steps: 358.15936

Negative log-marginal-likelihood after 11250/15625 steps: 144.10838

Negative log-marginal-likelihood after 11875/15625 steps: 89.537964

Negative log-marginal-likelihood after 12500/15625 steps: 59.200634

Negative log-marginal-likelihood after 13125/15625 steps: 46.43938

Negative log-marginal-likelihood after 13750/15625 steps: 40.257816

Negative log-marginal-likelihood after 14375/15625 steps: 35.257412

Negative log-marginal-likelihood after 15000/15625 steps: 26.825794

Negative log-marginal-likelihood after 15625/15625 steps: 11.478086

In [11]:

plt.plot(sol.t, data, "-", linewidth=5, alpha=0.5, label="Data")
tcoeffs = (u0, vf(u0, t=t0, p=f_args))
ibm, ssm = ivpsolvers.prior_ibm(tcoeffs, output_scale=1.0, ssm_fact="isotropic")
ts0 = ivpsolvers.correction_ts0(ssm=ssm)
strategy = ivpsolvers.strategy_smoother(ssm=ssm)
solver_ts0 = ivpsolvers.solver(strategy, prior=ibm, correction=ts0, ssm=ssm)
init = solver_ts0.initial_condition()

sol = ivpsolve.solve_fixed_grid(
    lambda *a, **kw: vf(*a, **kw, p=p), init, grid=grid, solver=solver_ts0, ssm=ssm
)


plt.plot(sol.t, sol.u[0], ".-", label="Final guess")

tcoeffs = (u0, vf(u0, t=t0, p=f_args))
ibm, ssm = ivpsolvers.prior_ibm(tcoeffs, output_scale=1.0, ssm_fact="isotropic")
ts0 = ivpsolvers.correction_ts0(ssm=ssm)
strategy = ivpsolvers.strategy_smoother(ssm=ssm)
solver_ts0 = ivpsolvers.solver(strategy, prior=ibm, correction=ts0, ssm=ssm)
init = solver_ts0.initial_condition()

sol = ivpsolve.solve_fixed_grid(
    lambda *a, **kw: vf(*a, **kw, p=f_args), init, grid=grid, solver=solver_ts0, ssm=ssm
)
plt.plot(sol.t, sol.u[0], ".-", label="Initial guess")


plt.legend()
plt.show()

plt.plot(sol.t, data, "-", linewidth=5, alpha=0.5, label="Data") tcoeffs = (u0, vf(u0, t=t0, p=f_args)) ibm, ssm = ivpsolvers.prior_ibm(tcoeffs, output_scale=1.0, ssm_fact="isotropic") ts0 = ivpsolvers.correction_ts0(ssm=ssm) strategy = ivpsolvers.strategy_smoother(ssm=ssm) solver_ts0 = ivpsolvers.solver(strategy, prior=ibm, correction=ts0, ssm=ssm) init = solver_ts0.initial_condition() sol = ivpsolve.solve_fixed_grid( lambda *a, **kw: vf(*a, **kw, p=p), init, grid=grid, solver=solver_ts0, ssm=ssm ) plt.plot(sol.t, sol.u[0], ".-", label="Final guess") tcoeffs = (u0, vf(u0, t=t0, p=f_args)) ibm, ssm = ivpsolvers.prior_ibm(tcoeffs, output_scale=1.0, ssm_fact="isotropic") ts0 = ivpsolvers.correction_ts0(ssm=ssm) strategy = ivpsolvers.strategy_smoother(ssm=ssm) solver_ts0 = ivpsolvers.solver(strategy, prior=ibm, correction=ts0, ssm=ssm) init = solver_ts0.initial_condition() sol = ivpsolve.solve_fixed_grid( lambda *a, **kw: vf(*a, **kw, p=f_args), init, grid=grid, solver=solver_ts0, ssm=ssm ) plt.plot(sol.t, sol.u[0], ".-", label="Initial guess") plt.legend() plt.show()

What's next¶

The same example can be constructed with deep learning libraries such as Equinox, Haiku, or Flax. To do so, define a corresponding vector field and a parameter set, build a new loss function and repeat.