Forward B-PINN: 1-D Poisson with noisy data

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Forward uncertainty quantification. Standard 1-D Poisson u''(x) = -sin(πx) with two noisy Dirichlet boundary observations. Instead of optax, the MLP weights are sampled via SGLD; after training, crux.eval([u]) auto-vmaps the evaluator over the chain to give a posterior mean and a credible band.

Why this matters

A deterministic PINN returns one weight vector and one prediction; you can't tell where the model is uncertain. A B-PINN's predictive band widens in data-sparse regions and tightens near observations — the mechanism Yang et al. 2021 illustrated as the central advantage of the Bayesian framework.

Reference

Yang, L., Meng, X., & Karniadakis, G. E. (2021). B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data. Journal of Computational Physics, 425, 109913.

Script

"""01 — Bayesian PINN forward problem: 1-D Poisson with noisy boundary data"""

from pathlib import Path

import blackjax
import foundax
import jax
import jax.numpy as jnp

import jno

π = jno.np.pi

# ── Domain ────────────────────────────────────────────────────────────────────
domain = jno.domain.line(mesh_size=0.1)
x, _ = domain.variable("interior")
xb, _ = domain.variable("boundary")

# ── Analytical solution and forcing ──────────────────────────────────────────
# u_exact(x) = sin(πx)/π² ⇒ u'' = -sin(πx).
u_exact_expr = jno.np.sin(π * x) / π**2
f_clean = -jno.np.sin(π * x)

# Sparse noisy sensor observations of u at the boundary.  jno.noise.gaussian
# draws a fresh observation each step from the solver's PRNG key — fully
# reproducible given the global seed.  (Yang et al. also noise the
# interior forcing sensors; that combination currently mixes
# interior/boundary point sets across constraints in jno's trace, so we
# noise only the boundary data here — sufficient to show the
# uncertainty-quantification benefit.)
sigma_b = 0.01
u_b_clean = jno.np.sin(π * xb) / π**2
u_b_obs = u_b_clean + jno.noise.gaussian(std=sigma_b)

# ── Bayesian PINN — SGLD posterior over MLP weights ───────────────────────────
u_net = jno.nn.wrap(
    foundax.mlp(
        in_features=1,
        hidden_dims=16,
        num_layers=2,
        key=jax.random.PRNGKey(0),
    )
)
u_net.bayesian(
    blackjax.sgld,
    step_size=1e-5,
    warmup=2000,
    keep=400,
    thin=2,
)

u = u_net(x)

# ── Constraints ───────────────────────────────────────────────────────────────
# PDE residual on noisy f-sensors, data fit on noisy boundary u-sensors.
pde = u.d2(x, scheme="finite_difference") - f_clean
bc = u_net(xb) - u_b_obs

# ── Solve ─────────────────────────────────────────────────────────────────────
crux = jno.core([pde.mse, bc.mse])
crux.solve(2800)

# ── Posterior prediction bands (auto-chain default) ──────────────────────────
u_chain = crux.eval([u])  # shape (K, N, n_points, 1) — K=1 by default
u_exact = crux.eval([u_exact_expr])  # no Bayesian deps → point value
# Reduce over both the chain (K) and sample (N) axes for per-point
# posterior summaries; with K=1 this is equivalent to axis=1 alone.
u_mean = jnp.mean(u_chain, axis=(0, 1))
u_lo, u_hi = jnp.quantile(u_chain, jnp.array([0.05, 0.95]), axis=(0, 1))

rel_l2 = float(jnp.linalg.norm(u_mean - u_exact) / (jnp.linalg.norm(u_exact) + 1e-8))
band_width = float(jnp.max(u_hi - u_lo))

results_file = Path(__file__).parent.parent.parent / "tutorial_results.txt"
with open(results_file, "a") as f:
    f.write(
        f"10_bayesian_pinns/01_forward_noisy_poisson_1d.py | epochs=2800 | "
        f"rel_L2_mean={rel_l2:.4f} | max_band_width={band_width:.4f}\n"
    )

# Loose tolerance: vanilla SGLD on a ~300-param MLP without
# preconditioning is RNG-path sensitive — the chain does not fully
# concentrate around the data-fit MAP in a tractable step budget.  The
# in-data rel-L2 is therefore noisy across reseeds.  For tightly
# calibrated bands the literature recommends preconditioned variants
# (pSGLD), SGHMC with mass-matrix adaptation, or variational inference.
# We only assert that the chain doesn't diverge and produces a
# non-trivial band — the qualitative B-PINN behaviour.
assert jnp.isfinite(rel_l2), f"posterior mean diverged: rel_l2 = {rel_l2}"
assert rel_l2 < 100.0, f"posterior-mean rel L2 unreasonably high: {rel_l2:.3e}"
assert band_width > 1e-4, f"credible band collapsed: max width {band_width:.3e}"