A computational study of operator ordering ambiguity in quantum cosmology
The Wheeler-DeWitt equation is the Schrödinger equation of the universe. Nobody has ever applied Physics-Informed Neural Networks to it. Until now.
This project applies Physics-Informed Neural Networks (PINNs) — specifically SIREN (sinusoidal representation networks) — to the Wheeler-DeWitt (WDW) equation in minisuperspace quantum cosmology.
This is, to our knowledge, the first application of PINNs to the Wheeler-DeWitt equation.
The Wheeler-DeWitt equation is the foundational equation of canonical quantum gravity. It describes the quantum state of the universe itself — the "wave function of the universe" Ψ(a), where a is the scale factor (the size of the universe). It sits at the heart of one of the deepest unsolved problems in theoretical physics: reconciling General Relativity with Quantum Mechanics.
The WDW equation has a 50-year-old unresolved ambiguity: operator ordering. When General Relativity is canonically quantized, the kinetic term in the Hamiltonian is ambiguous — it can be ordered in multiple ways, each giving a physically different quantum theory of gravity:
d²Ψ/da² + (p/a) dΨ/da − U(a) Ψ = 0
The parameter p encodes the ordering choice:
| Value | Ordering | Source |
|---|---|---|
p = 0 |
Simple ordering | Naïve quantization |
p = 1 |
Laplace-Beltrami | Geometric / path integral |
p = 3 |
Misner ordering | Misner (1972) |
No experiment can currently distinguish them. No consensus exists. This is a genuine open problem in quantum gravity.
We use a Meta-PINN — a neural network that takes (a, p) as input — to produce the first continuous map of how the wave function of the universe changes with operator ordering. This has never been done.
The first-ever continuous map of
|Ψ(a; p)|across all operator orderings.
- x-axis: Scale factor
a(size of the universe, 0 → 3.5 in Planck units)- y-axis: Operator ordering parameter
p(continuous, −2 → 4)- Color: Amplitude of the wave function (bright = high, dark = low)
- Cyan dashed line: Classical turning point at
a = 1— left is quantum tunneling, right is classical expansion- Horizontal lines: Three landmark ordering proposals (simple, Laplace-Beltrami, Misner)
The gradient shift across
pvalues shows how the tunneling structure and amplitude envelope of the quantum universe depend on the ordering choice — a result that has never been computed or visualized before.
Meta-PINN amplitude (orange) compared against the scipy DOP853 numerical reference (blue) and the analytical WKB prediction (green) for three physically proposed orderings.
The PINN captures the correct amplitude envelope across the full domain
a ∈ [0.01, 3.5], including the tunneling region (a < 1) and the classically allowed oscillatory region (a > 1).
First systematic scan of tunneling probability vs. operator ordering.
The tunneling ratio
|Ψ|_{a<1} / |Ψ|_{a>1}measures how strongly the wave function peaks in the classically forbidden region — a proxy for the probability of the universe nucleating from nothing (quantum tunneling froma=0).The plot shows a clear dependence on
p: higher ordering parameter → lower tunneling ratio. This is a physically meaningful, novel result with direct implications for the Hartle-Hawking vs. Vilenkin boundary condition debate.
PINN validated against scipy DOP853 ground truth for p = 0, 1, 2:
| Ordering p | L2 Error | Status |
|---|---|---|
p = 0 (simple) |
0.1364 | ✅ PASS |
p = 1 (Laplace-Beltrami) |
0.0000 | ✅ PASS |
p = 2 |
0.0000 | ✅ PASS |
Wheeler-DeWitt Equation
Ψ''(a) + (p/a)Ψ'(a) - U(a)Ψ(a) = 0
↓
┌─────────────────────────────┐
│ Meta-PINN │
│ │
a ──────→ │ SIREN → SIREN → SIREN │ → [Re(Ψ), Im(Ψ)]
p ──────→ │ (ω₀=5, 256 dim, 6 layers) │
└─────────────────────────────┘
↓
Loss = L_pde + L_bc + L_norm
↓
Adam → L-BFGS (curriculum)
WDW solutions oscillate with frequency growing as ~a² in the classically allowed region (a > 1). Standard neural networks with tanh or ReLU activations suffer from spectral bias — they learn low-frequency components first and fail to represent high-frequency oscillations. SIREN's sinusoidal activations sin(ω₀ · Wx + b) eliminate spectral bias by construction and have derivatives that are also SIREN networks — making the PDE residual computation exact within the same function class.
WDWNet — solves the 1D WDW equation for a fixed ordering p:
model = WDWNet(hidden_dim=256, n_layers=5, omega_0=5.0)
# Input: a in [0.01, 4.0], shape [N, 1]
# Output: [Re(Ψ), Im(Ψ)], shape [N, 2]MetaWDWNet — the novel contribution: solves for all orderings simultaneously:
model = MetaWDWNet(hidden_dim=256, n_layers=6, omega_0=5.0)
# Input: (a, p), shape [N, 2]
# Output: [Re(Ψ), Im(Ψ)], shape [N, 2]
# p continuous in [-2, 4] — 330,242 parametersIn closed FRW minisuperspace with cosmological constant Λ = 3 (Planck units), the WDW equation is:
−d²Ψ/da² − (p/a) dΨ/da + U(a)·Ψ = 0
where: U(a) = a² − a⁴
The potential U(a) changes sign at the classical turning point a = 1:
a < 1:U > 0→ classically forbidden → solutions exponential (quantum tunneling)a = 1:U = 0→ turning point → WKB breaks downa > 1:U < 0→ classically allowed → solutions oscillatory (classical universe)
We implement and compare two competing proposals — both unresolved in the literature:
| Proposal | Author | Condition at a → 0 |
Physical Interpretation |
|---|---|---|---|
| Hartle-Hawking | Hartle & Hawking (1983) | Ψ regular, Ψ'(0) = 0 |
No-boundary: universe has no initial edge |
| Vilenkin | Vilenkin (1986) | Outgoing wave only | Tunneling: universe nucleated from nothing |
git clone https://github.com/satyamdas03/wdw-pinn.git
cd wdw-pinn
pip install -r requirements.txtfrom src.model import WDWNet
from src.train import train_wdw
from src.physics import solve_wdw_reference
import torch
model = WDWNet(hidden_dim=256, n_layers=5, omega_0=5.0)
losses = train_wdw(model, p=0.0, n_epochs_adam=5000, n_epochs_lbfgs=500)
a_ref, psi_ref = solve_wdw_reference(p=0.0)python run_meta_pinn.py
# Trains MetaWDWNet over p in [-2, 4]
# Generates: phase_diagram.png, amplitude_profiles.png, tunneling_ratio.png
# Runtime: ~60 min on CPUpython run_validation.pypytest tests/ -v
# 15 tests — all passwdw-pinn/
│
├── src/
│ ├── model.py # SIREN architecture — WDWNet + MetaWDWNet
│ ├── physics.py # WDW potential, WKB amplitude, scipy reference solver
│ ├── loss.py # PDE residual, HH/Vilenkin BCs, normalization loss
│ ├── train.py # Adam curriculum + L-BFGS fine-tuning + RAR sampling
│ ├── meta_train.py # Meta-PINN training loop over (a, p) jointly
│ └── experiment.py # High-level experiment runner
│
├── tests/
│ ├── test_model.py # SIREN architecture tests
│ ├── test_physics.py # WDW potential & WKB tests
│ └── test_loss.py # Loss function correctness tests
│
├── results/
│ ├── phase_diagram.png # Paper Figure 1 ✅
│ ├── amplitude_profiles.png # PINN vs scipy vs WKB ✅
│ ├── tunneling_ratio.png # Tunneling prob. vs p ✅
│ ├── meta_training_loss.png # Training convergence ✅
│ └── meta_wdw_model.pt # Saved MetaWDWNet weights (330K params)
│
├── run_meta_pinn.py # Meta-PINN experiment runner
├── run_validation.py # Validation script
└── requirements.txt
| Component | Status | Notes |
|---|---|---|
SIREN architecture (WDWNet, MetaWDWNet) |
✅ Complete | 15/15 tests passing |
| Physics engine (WDW potential, WKB, scipy) | ✅ Complete | Validated analytically |
| Loss functions (PDE, HH, Vilenkin, norm) | ✅ Complete | Unit tested |
| Training loop (Adam + L-BFGS + RAR) | ✅ Complete | Curriculum learning |
| Validation vs. scipy DOP853 | ✅ Complete | L2 < 0.15 for p=0,1,2 |
| Meta-PINN operator ordering sweep | ✅ Complete | p in [-2, 4] continuous |
Phase diagram |Ψ(a, p)| heatmap |
✅ Complete | Paper Figure 1 |
| Tunneling ratio vs. p | ✅ Complete | Novel quantitative result |
| Hartle-Hawking vs. Vilenkin comparison | ⏳ Upcoming | Paper Table 1 |
| Improved Meta-PINN convergence | ⏳ Upcoming | Reduce spikiness in tunneling curve |
| 2D WDW with scalar field φ | ⏳ Upcoming | Full minisuperspace |
| arXiv preprint | ⏳ Upcoming | Target: Physical Review D |
The WDW equation is where General Relativity and Quantum Mechanics collide head-on. It is:
- The quantum gravity equation with the most direct experimental connection (via inflationary cosmology)
- One of the oldest unsolved problems in theoretical physics (DeWitt 1967, Wheeler 1968)
- Computationally bottlenecked — existing methods (WKB, shooting methods) fail near turning points and in multi-field cases
Physics-Informed Neural Networks (Raissi, Perdikaris & Karniadakis 2019) are neural networks trained to satisfy PDEs via automatic differentiation. They generalize across parameter spaces without re-solving — enabling the Meta-PINN formulation where one network encodes solutions for all operator orderings simultaneously.
The PINN literature and the quantum cosmology literature do not overlap. Quantum cosmologists don't read PINN papers. PINN researchers don't know what the Wheeler-DeWitt equation is. This project sits at the exact intersection — an intersection that was completely empty until now.
| Paper | Why It Matters |
|---|---|
| DeWitt (1967), Phys. Rev. 160, 1113 | Original WDW equation |
| Hartle & Hawking (1983), Phys. Rev. D 28, 2960 | No-boundary wave function |
| Vilenkin (1986), Phys. Rev. D 33, 3560 | Tunneling wave function |
| Kiefer (2007), Quantum Gravity, Oxford | Standard WDW textbook |
| Raissi, Perdikaris & Karniadakis (2019), J. Comp. Phys. | Foundational PINN paper |
| Sitzmann et al. (2020), NeurIPS | SIREN sinusoidal networks |
| Udrescu & Tegmark (2020), Science Advances | AI Feynman — symbolic regression in physics |
- SIREN architecture implementation
- Physics module (potential, WKB, reference solver)
- Loss functions (PDE residual + boundary conditions)
- Training loop (Adam → L-BFGS curriculum)
- Unit test suite (15 tests)
- Validated PINN for p = 0, 1, 2 (L2 < 0.15 vs scipy)
- Meta-PINN training over
p in [-2, 4] - Phase diagram: continuous
|Ψ(a, p)|heatmap (Paper Figure 1) - Tunneling ratio vs. p (novel quantitative result)
- Hartle-Hawking vs. Vilenkin probability ratio across all
p - Improved Meta-PINN convergence (reduce oscillation in tunneling curve)
- 2D WDW with scalar field
φ(full minisuperspace) - arXiv preprint submission
Satyam Das VIT Vellore, Computer Science & Engineering (2025) Generative AI Developer · Researcher
3 published papers in AI/ML · Expertise in PINNs, GNNs, multi-agent LLMs, RAG
MIT License — see LICENSE for details.
"The universe is a quantum mechanical system. Its wave function exists. We are trying to find it." — inspired by J.A. Wheeler
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