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Getting Started

This page walks you through running your first example and understanding the core jNO workflow.

Before you begin, complete the setup steps in Installation.

Running Your First Example

cd examples
uv run python laplace1D.py

This solves the 1D Laplace equation ∂²u/∂x² = sin(πx) with a Physics-Informed Neural Network (PINN) and saves results to ./runs/laplace1D/.

Core Workflow

Every jNO program follows this five-step pattern:

1. Define the Domain

import jno

# 2D rectangular domain with mesh spacing 0.05
domain = jno.domain(constructor=jno.domain.rect(mesh_size=0.05))

See Domain & Geometry for all supported geometries.

2. Sample Variables

x, y, t = domain.variable("interior")    # interior collocation points
xb, yb, tb = domain.variable("boundary") # boundary points

3. Define the Neural Network

import jno.numpy as jnn
import jax

u_net = jnn.nn.mlp(in_features=2, hidden_dims=64, num_layers=3, key=jax.random.PRNGKey(0))
u = u_net(x, y)

4. Formulate Constraints

import jno.numpy as jnn

π = jnn.pi
pde = -jnn.laplacian(u, [x, y]) - 1.0   # ∇²u + 1 = 0
boc = u(xb, yb) - 0.0                    # u=0 on boundary

Constraints are symbolic expressions — no computation happens at this stage.

5. Solve

import optax
from jno import LearningRateSchedule as lrs

u_net.optimizer(optax.adam, lr=lrs.exponential(1e-3, 0.9, 2000, 1e-5))
crux = jno.core([pde.mse, boc.mse], domain)
stats = crux.solve(2000)
stats.plot("history.png")

Project Setup Helper

jno.setup() initialises logging and returns the run directory in one call:

dire = jno.setup(__file__)   # creates ./runs/<script_name>/
dire = jno.setup(__file__, name="experiment_v1")  # custom name

Understanding the Output

During training jNO prints a progress line each epoch:

Epoch  1000/2000| L: 1.2345e-03 | C0: 1.1000e-03 | C1: 1.3500e-04
  • L — total weighted loss
  • C0, C1 — individual constraint losses
  • T0, T1 — tracker values (if any)

After training, stats.plot("history.png") saves the loss curves.

Next Steps

Topic Page
All geometry types Domain & Geometry
Neural operator architectures Neural Network Architectures
Differential operators Differential Operators
Optimizers, LR schedules, constraint weights Training
Residual-adaptive point selection Adaptive Resampling
Architecture and hyperparameter search Hyperparameter Tuning
Saving and loading solvers Save, Load & Configuration