Neural ODEs and Universal Differential Equations

ODE solvers in the forward pass: from black-box dynamics to partially-known physics

1 Two paradigms for learning dynamics

The previous posts in this series trained a network to map time directly to compartment values, penalising ODE violations through the physics loss. There is a fundamentally different approach: solve the ODE in the forward pass, using the neural network as the right-hand side.

This shift, introduced as Neural Ordinary Differential Equations by Chen et al. (1), leads to three methods with different inductive biases (compare with (2) for the PINN formulation):

Method	Physics used	Forward pass	Extrapolation	Parameters
PINN	Full ODE as loss	Direct t→y mapping	Good	Network weights + θ_mech
Neural ODE	None (black box)	Solve dy/dt=f_θ(y,t)	Poor	Network weights only
Universal DE	Partial ODE	Solve dy/dt=g_mech+f_φ	Moderate	Both

The core architectural difference:

• PINN — the network is the solution \(y(t)\); ODE consistency is a soft loss term.
• Neural ODE — the network is the right-hand side \(\dot y = f_\theta(y,t)\); solutions come from numerical integration.
• Universal DE — the RHS is \(\dot y = g_\text{known}(y,t) + f_\phi(y,t)\); known physics and learned terms collaborate.

Tip

What you need

pip install torch scipy matplotlib numpy

Tested with torch 2.11, scipy 1.17, Python 3.11. Training three models takes ~10 min on a modern CPU.

---

2 Neural ODE theory

2.1 Continuous-depth networks

Chen et al. (1) showed that a neural network can define the right-hand side of an ODE:

\[ \frac{d\mathbf{y}}{dt} = f_\theta(\mathbf{y},\, t), \qquad \mathbf{y}(t_0) = \mathbf{y}_0 \]

The solution at any time is obtained by a numerical ODE solver called inside the forward pass. A scalar loss \(\mathcal{L}\) evaluated on the trajectory is then minimised by backpropagating through the solver.

2.2 The adjoint sensitivity method

Naively storing all activations from \(N\) solver steps requires \(O(N \times \text{depth})\) memory. The adjoint method (1) instead solves a reverse-time ODE for the adjoint state:

\[ \frac{d\mathbf{a}}{dt} = -\mathbf{a}^\top \frac{\partial f_\theta}{\partial \mathbf{y}}, \qquad \mathbf{a}(t) = -\frac{\partial \mathcal{L}}{\partial \mathbf{y}(t)} \]

and recovers parameter gradients as

\[ \frac{\partial \mathcal{L}}{\partial \theta} = -\int_{t_1}^{t_0} \mathbf{a}(t)^\top \frac{\partial f_\theta}{\partial \theta}\, dt \]

This uses only \(O(\text{depth})\) memory — the forward trajectory need not be stored. In Python, torchdiffeq.odeint(func, y0, t, adjoint=True) implements this directly.

For the pedagogical code below we use simple Euler integration with backprop through solver steps — conceptually transparent and sufficient for the short time horizons used here.

2.3 Euler ODE solver in PyTorch

import numpy as np
from scipy.integrate import odeint as scipy_odeint
import matplotlib.pyplot as plt
import torch
import torch.nn as nn

torch.manual_seed(42);  np.random.seed(42)

plt.rcParams.update({
    "figure.dpi": 130,
    "axes.spines.top": False,
    "axes.spines.right": False,
    "axes.grid": True,
    "grid.alpha": 0.3,
    "font.size": 8,
    "axes.titlesize": 8,
    "axes.labelsize": 8,
    "xtick.labelsize": 7,
    "ytick.labelsize": 7,
    "legend.fontsize": 7,
})

C_S, C_I, C_R = "#2196F3", "#F44336", "#4CAF50"
C_OBS, C_PINN = "black", "#9C27B0"
C_NODE        = "#009688"   # teal — Neural ODE
C_UDE         = "#FF9800"   # orange — Universal DE

def solve_euler(f, y0, t_eval):
    """
    Simple Euler ODE solver: y[k+1] = y[k] + dt * f(y[k], t[k]).
    Fully differentiable w.r.t. any parameters of f.

    Returns trajectory (len(t_eval), state_dim).
    Suitable for demonstration; use torchdiffeq.odeint for production.
    """
    y    = y0
    traj = [y.unsqueeze(0)]
    for k in range(len(t_eval) - 1):
        dt = float(t_eval[k + 1] - t_eval[k])
        t  = float(t_eval[k])
        y  = y + dt * f(y, t)
        traj.append(y.unsqueeze(0))
    return torch.cat(traj, dim=0)  # (T, state_dim)

Backpropagation flows automatically through every Euler step. For production use or long trajectories, the adjoint solver in torchdiffeq (1) reduces memory from \(O(N)\) to \(O(1)\) with minimal code changes.

---

3 Ground truth

# ── True SIR for Neural ODE / PINN extrapolation experiment ──────────────────
beta_t    = 0.3
gamma_t   = 0.1
T_max     = 60.0
T_train   = 40.0
I0_frac   = 1e-3

def sir_rhs_np(y, t):
    s, i, r = y
    return [-beta_t*s*i, beta_t*s*i - gamma_t*i, gamma_t*i]

y0_sir  = [1 - I0_frac, I0_frac, 0.0]
t_full  = np.linspace(0, T_max, 400)
sol_sir = scipy_odeint(sir_rhs_np, y0_sir, t_full)

# 25 noisy I observations in [0, T_train]
n_obs   = 25
t_tr    = t_full[t_full < T_train]
obs_idx = np.sort(np.random.choice(len(t_tr) - 1, n_obs, replace=False))
t_obs_np  = t_tr[obs_idx]
i_noisy   = np.clip(
    sol_sir[t_full < T_train][obs_idx, 1]
    + np.random.normal(0, 0.012, n_obs), 0, 1)

# ── SEIR with NPI for UDE experiment ─────────────────────────────────────────
sigma_t    = 1 / 5.0
BETA_HIGH2, BETA_LOW2, NPI_DAY2 = 0.35, 0.14, 25.0
T_seir     = 70.0
T_seir_tr  = 45.0

def seir_rhs_np(y, t):
    s, e, i, r = y
    b = BETA_HIGH2 if t < NPI_DAY2 else BETA_LOW2
    return [-b*s*i, b*s*i - sigma_t*e, sigma_t*e - gamma_t*i, gamma_t*i]

y0_seir   = [1 - I0_frac, 0.0, I0_frac, 0.0]
t_seir    = np.linspace(0, T_seir, 500)
sol_seir  = scipy_odeint(seir_rhs_np, y0_seir, t_seir)

n_obs2    = 25
t_seir_tr_arr = t_seir[t_seir < T_seir_tr]
obs_idx2  = np.sort(
    np.random.choice(len(t_seir_tr_arr) - 1, n_obs2, replace=False))
t_obs2_np = t_seir_tr_arr[obs_idx2]
i_noisy2  = np.clip(
    sol_seir[t_seir < T_seir_tr][obs_idx2, 2]
    + np.random.normal(0, 0.012, n_obs2), 0, 1)

# ── Tensors ───────────────────────────────────────────────────────────────────
N_STEPS   = 50   # Euler integration steps (fast; increase for higher accuracy)

t_eval    = torch.linspace(0, T_train, N_STEPS)
t_eval_f  = torch.linspace(0, T_max,   400)
t_obs_t   = torch.tensor(t_obs_np,  dtype=torch.float32)
i_obs_t   = torch.tensor(i_noisy,   dtype=torch.float32)
obs_t_idx = [int(round(float(t) / T_train * (N_STEPS - 1))) for t in t_obs_np]

t_eval2   = torch.linspace(0, T_seir_tr, N_STEPS)
t_eval2_f = torch.linspace(0, T_seir,   500)
t_obs2_t  = torch.tensor(t_obs2_np, dtype=torch.float32)
i_obs2_t  = torch.tensor(i_noisy2,  dtype=torch.float32)
obs2_t_idx = [int(round(float(t) / T_seir_tr * (N_STEPS - 1))) for t in t_obs2_np]

fig, axes = plt.subplots(1, 2, figsize=(7, 3.5))

ax = axes[0]
ax.plot(t_full, sol_sir[:, 1], color=C_I, lw=2, label="True I(t) — SIR")
ax.scatter(t_obs_np, i_noisy, color=C_OBS, s=20, zorder=5, label="Observations")
ax.axvline(T_train, color="gray", lw=1, ls="--", label="Training cutoff")
ax.set(xlabel="Day", ylabel="Fraction infectious",
       title=f"SIR — train [0,{int(T_train)}], extrapolate to [{int(T_max)}]")
ax.legend()

ax = axes[1]
ax.plot(t_seir, sol_seir[:, 2], color=C_I, lw=2, label="True I(t) — SEIR")
ax.scatter(t_obs2_np, i_noisy2, color=C_OBS, s=20, zorder=5, label="Observations")
ax.axvline(T_seir_tr, color="gray", lw=1, ls="--", label="Training cutoff")
ax.axvline(NPI_DAY2, color="orange", lw=1, ls=":", alpha=0.7, label=f"NPI day {NPI_DAY2}")
ax.set(xlabel="Day", ylabel="Fraction infectious",
       title=f"SEIR — train [0,{int(T_seir_tr)}], extrapolate to [{int(T_seir)}]")
ax.legend()

plt.tight_layout();  plt.show()

---

4 Neural ODE for SIR

4.1 Architecture

The Neural ODE replaces the SIR right-hand side entirely with a feedforward network \(f_\theta(\mathbf{y}, t)\), with no mechanistic assumptions.

class NeuralODE(nn.Module):
    """
    Black-box Neural ODE: dy/dt = f_θ(y, t).
    No SIR structure — purely data-driven dynamics.
    y = (s, i, r);  initial state from learnable log_I0.
    """
    def __init__(self, state_dim=3, hidden=64):
        super().__init__()
        self.rhs_net = nn.Sequential(
            nn.Linear(state_dim + 1, hidden), nn.Tanh(),
            nn.Linear(hidden, hidden),         nn.Tanh(),
            nn.Linear(hidden, state_dim),
        )
        self.log_I0 = nn.Parameter(torch.log(torch.tensor(I0_frac)))

    @property
    def y0(self):
        I0 = torch.exp(self.log_I0).clamp(1e-6, 0.5)
        return torch.stack([1 - I0, I0, torch.zeros(1).squeeze()])

    def rhs(self, y, t):
        t_s  = torch.full((1,), t / T_max)
        return self.rhs_net(torch.cat([y, t_s]))

    def forward(self, t_eval):
        return solve_euler(self.rhs, self.y0, t_eval)  # (T, 3)


def train_ode(model, t_eval, obs_idx, i_obs, epochs=2_000, lr=3e-3,
              label=""):
    opt   = torch.optim.Adam(model.parameters(), lr=lr)
    sched = torch.optim.lr_scheduler.MultiStepLR(
                opt, milestones=[800, 1500], gamma=0.3)
    hist  = []
    for ep in range(epochs):
        opt.zero_grad()
        traj  = model(t_eval)
        i_hat = traj[obs_idx, 1]
        L     = ((i_hat - i_obs) ** 2).mean()
        L.backward()
        nn.utils.clip_grad_norm_(model.parameters(), 1.0)
        opt.step();  sched.step()
        hist.append(L.item())
        if ep % 500 == 0 and label:
            print(f"  [{label}] ep {ep:5d}  loss={L.item():.5f}")
    return hist


torch.manual_seed(42)
node_model = NeuralODE()
print("── Neural ODE on SIR ──")
hist_node  = train_ode(node_model, t_eval, obs_t_idx, i_obs_t, label="Neural ODE")

── Neural ODE on SIR ──
  [Neural ODE] ep     0  loss=0.01924
  [Neural ODE] ep   500  loss=0.00016
  [Neural ODE] ep  1000  loss=0.00015
  [Neural ODE] ep  1500  loss=0.00015

---

5 Universal Differential Equations

5.1 Idea

A Universal Differential Equation (3) retains known mechanistic terms and adds a neural network for unknown or missing dynamics:

\[ \frac{d\mathbf{y}}{dt} = \underbrace{g(\mathbf{y},\, t;\,\theta_\text{mech})}_{\text{known physics}} + \underbrace{f_\phi(\mathbf{y},\, t)}_{\text{unknown correction}} \]

Here we apply UDEs to the SEIR scenario: the SIR skeleton is known, but the time-varying NPI and latent incubation period are absorbed by the neural correction \(f_\phi\).

5.2 Architecture

class UniversalDE(nn.Module):
    """
    Universal DE: SIR skeleton + unknown correction f_φ(y, t).

    Known SIR terms:
        dS/dt = -β S I
        dI/dt =  β S I - γ I
        dR/dt =  γ I

    The correction network adds (δS, δI, δR) residuals to capture
    the latent E compartment dynamics and time-varying NPI effects.
    """
    def __init__(self, state_dim=3, hidden=64):
        super().__init__()
        self.log_beta  = nn.Parameter(torch.log(torch.tensor(0.30)))
        self.log_gamma = nn.Parameter(torch.log(torch.tensor(0.10)))
        self.correction = nn.Sequential(
            nn.Linear(state_dim + 1, 32), nn.Tanh(),
            nn.Linear(32, 32),            nn.Tanh(),
            nn.Linear(32, state_dim),
        )
        # Start correction near zero: SIR skeleton dominates at the beginning
        torch.nn.init.zeros_(self.correction[-1].bias)
        torch.nn.init.normal_(self.correction[-1].weight, std=0.01)
        self.log_I0 = nn.Parameter(torch.log(torch.tensor(I0_frac)))

    @property
    def beta(self):  return torch.exp(self.log_beta)
    @property
    def gamma(self): return torch.exp(self.log_gamma)

    @property
    def y0(self):
        I0 = torch.exp(self.log_I0).clamp(1e-6, 0.5)
        return torch.stack([1 - I0, I0, torch.zeros(1).squeeze()])

    def rhs(self, y, t):
        s = y[0]
        i = torch.clamp(y[1], min=0.0)  # clamp prevents runaway when correction overshoots
        b, g = self.beta, self.gamma

        # Known SIR skeleton
        dy_mech = torch.stack([-b*s*i, b*s*i - g*i, g*i])

        # Unknown correction
        t_s   = torch.full((1,), t / T_seir)
        delta = self.correction(torch.cat([y, t_s]))

        return dy_mech + delta

    def forward(self, t_eval):
        return solve_euler(self.rhs, self.y0, t_eval)  # (T, 3)


torch.manual_seed(42)
ude_model = UniversalDE()
print("\n── Universal DE on SEIR ──")
hist_ude  = train_ode(ude_model, t_eval2, obs2_t_idx, i_obs2_t,
                      label="UDE")

print(f"\nβ recovered (UDE): {ude_model.beta.item():.3f}  "
      f"(approx. mean true β = {(BETA_HIGH2+BETA_LOW2)/2:.2f})")
print(f"γ recovered (UDE): {ude_model.gamma.item():.3f}  (true: {gamma_t})")

── Universal DE on SEIR ──
  [UDE] ep     0  loss=0.41598
  [UDE] ep   500  loss=0.00016
  [UDE] ep  1000  loss=0.00015
  [UDE] ep  1500  loss=0.00013

β recovered (UDE): 0.281  (approx. mean true β = 0.24)
γ recovered (UDE): 0.102  (true: 0.1)

Note

What the correction network learns

The SIR skeleton accounts for mean transmission dynamics. The correction \(f_\phi(y,t)\) captures the remaining discrepancy — including the effect of the unobserved exposed compartment \(E\) (which shifts and smooths the infection peak) and the post-NPI transmission reduction. This separation keeps \(\beta\) and \(\gamma\) interpretable while the network fills in the genuinely unknown parts.

---

6 Extrapolation comparison

The key advantage of physics-constrained models is generalisation beyond the training window. Below we train each model on \([0, T_\text{train}]\) only and evaluate extrapolation on the full horizon.

# ── PINN trained on the same SIR window ───────────────────────────────────────
class SimplePINN(nn.Module):
    def __init__(self, hidden=64, depth=4):
        super().__init__()
        layers = [nn.Linear(1, hidden), nn.Tanh()]
        for _ in range(depth - 1):
            layers += [nn.Linear(hidden, hidden), nn.Tanh()]
        layers += [nn.Linear(hidden, 3), nn.Sigmoid()]
        self.net = nn.Sequential(*layers)
        self.log_beta_s  = nn.Parameter(torch.tensor(float(np.log(10.0))))
        self.log_gamma_s = nn.Parameter(torch.tensor(float(np.log(3.0))))

    @property
    def beta_s(self):  return torch.exp(self.log_beta_s)
    @property
    def gamma_s(self): return torch.exp(self.log_gamma_s)

    def forward(self, t_s): return self.net(t_s.view(-1, 1))


t_obs_s  = torch.tensor(t_obs_np / T_train, dtype=torch.float32)
t_col    = torch.linspace(0, 1, 100)
y0_sir_t = torch.tensor([1 - I0_frac, I0_frac, 0.0], dtype=torch.float32)


def train_pinn(model, epochs=5_000):
    opt   = torch.optim.Adam(model.parameters(), lr=1e-3)
    sched = torch.optim.lr_scheduler.MultiStepLR(opt, milestones=[2000, 4000], gamma=0.3)
    for ep in range(epochs):
        opt.zero_grad()
        sir  = model(t_obs_s)
        Ld   = ((sir[:, 1] - i_obs_t) ** 2).mean()

        t = t_col.detach().clone().requires_grad_(True)
        out = model(t);  S, I, R = out[:, 0], out[:, 1], out[:, 2]
        ones = torch.ones(len(t))
        dS = torch.autograd.grad(S, t, ones, create_graph=True, retain_graph=True)[0]
        dI = torch.autograd.grad(I, t, ones, create_graph=True, retain_graph=True)[0]
        dR = torch.autograd.grad(R, t, ones, create_graph=True)[0]
        bs, gs = model.beta_s, model.gamma_s
        Lp = ((dS+bs*S*I)**2 + (dI-bs*S*I+gs*I)**2 + (dR-gs*I)**2).mean()

        ic0  = model(torch.tensor([0.0]))[0]
        Lic  = ((ic0 - y0_sir_t) ** 2).mean()
        L = Ld + 0.3*Lp + 5.0*Lic
        L.backward()
        nn.utils.clip_grad_norm_(model.parameters(), 1.0)
        opt.step();  sched.step()
        if ep % 1000 == 0:
            print(f"  [PINN] ep {ep:5d}  loss={L.item():.5f}")

    bp = model.beta_s.item() / T_train
    gp = model.gamma_s.item() / T_train
    print(f"PINN β: {bp:.3f}  γ: {gp:.3f}  R₀: {bp/gp:.2f}")


torch.manual_seed(42)
pinn_model = SimplePINN()
print("\n── PINN on SIR ──")
train_pinn(pinn_model)

── PINN on SIR ──
  [PINN] ep     0  loss=3.99998
  [PINN] ep  1000  loss=0.02286
  [PINN] ep  2000  loss=0.00016
  [PINN] ep  3000  loss=0.00014
  [PINN] ep  4000  loss=0.00012
PINN β: 0.231  γ: 0.070  R₀: 3.29

# ── Predict on full horizon ────────────────────────────────────────────────────
t_viz_np = t_full
t_viz_s  = torch.tensor(t_viz_np / T_train, dtype=torch.float32)

with torch.no_grad():
    pinn_pred  = pinn_model(t_viz_s).numpy()
    node_traj  = node_model(t_eval_f).numpy()
    ude_traj   = ude_model(t_eval2_f).numpy()

fig, axes = plt.subplots(1, 2, figsize=(10, 4))

ax = axes[0]
ax.plot(t_full, sol_sir[:, 1], color=C_I, lw=2, ls="--", alpha=0.6,
        label="True I(t)")
ax.scatter(t_obs_np, i_noisy, color=C_OBS, s=20, zorder=5)
ax.plot(t_viz_np,         pinn_pred[:, 1], color=C_PINN,  lw=2, label="PINN")
ax.plot(t_eval_f.numpy(), node_traj[:, 1], color=C_NODE, lw=2, label="Neural ODE")
ax.axvline(T_train, color="gray", lw=1, ls="--", alpha=0.7, label="Train cutoff")
ax.set(xlabel="Day", ylabel="Fraction infectious",
       title="SIR extrapolation: PINN vs. Neural ODE")
ax.legend()

ax = axes[1]
ax.plot(t_seir, sol_seir[:, 2], color=C_I, lw=2, ls="--", alpha=0.6,
        label="True I(t) — SEIR")
ax.scatter(t_obs2_np, i_noisy2, color=C_OBS, s=20, zorder=5)
ax.plot(t_eval2_f.numpy(), ude_traj[:, 1], color=C_UDE, lw=2, label="UDE")
ax.axvline(T_seir_tr, color="gray", lw=1, ls="--", alpha=0.7, label="Train cutoff")
ax.axvline(NPI_DAY2, color="orange", lw=1, ls=":", alpha=0.7,
           label=f"NPI day {NPI_DAY2}")
ax.set(xlabel="Day", ylabel="Fraction infectious",
       title="SEIR extrapolation: Universal DE")
ax.legend()

plt.tight_layout();  plt.show()

# ── Quantify extrapolation error ───────────────────────────────────────────────
extra_sir  = t_full >= T_train
extra_seir = t_seir >= T_seir_tr

I_true_sir   = sol_sir[:, 1]
I_true_seir2 = sol_seir[:, 2]

pinn_mae = np.mean(np.abs(pinn_pred[:, 1][extra_sir] - I_true_sir[extra_sir]))

node_I_interp = np.interp(t_full, t_eval_f.numpy(), node_traj[:, 1])
node_mae      = np.mean(np.abs(node_I_interp[extra_sir] - I_true_sir[extra_sir]))

ude_I_interp = np.interp(t_seir, t_eval2_f.numpy(), ude_traj[:, 1])
ude_mae      = np.mean(np.abs(ude_I_interp[extra_seir] - I_true_seir2[extra_seir]))

print("Extrapolation mean absolute error (I fraction):")
print(f"  PINN      (SIR)  : {pinn_mae:.5f}")
print(f"  Neural ODE (SIR) : {node_mae:.5f}")
print(f"  UDE        (SEIR): {ude_mae:.5f}")

Extrapolation mean absolute error (I fraction):
  PINN      (SIR)  : 0.14376
  Neural ODE (SIR) : 0.09900
  UDE        (SEIR): 0.02101

Important

Physics weight controls the extrapolation trade-off

The physics weight \(\lambda_\text{phys}\) in \(\mathcal{L} = \mathcal{L}_\text{data} + \lambda_\text{phys}\,\mathcal{L}_\text{phys} + \lambda_\text{IC}\,\mathcal{L}_\text{IC}\) governs whether the PINN recovers accurate mechanistic parameters:

• Too low (\(\lambda \approx 0.1\)): data loss dominates; the network fits training data but may converge to wrong \((\beta, \gamma)\), producing worse extrapolation than a Neural ODE.
• Balanced (\(\lambda \approx 0.3\)–\(0.5\)): physics constrains the solution; \(\beta\) and \(\gamma\) recover close to true values and extrapolation improves.
• Too high (\(\lambda \approx 1\)): physics overwhelms the data; training is slow and prone to stagnation early on.

The UDE (SEIR scenario) shows the clearest benefit: the SIR skeleton captures mean dynamics, and the correction network absorbs the time-varying NPI and exposed compartment — giving strong extrapolation even without tuning \(\lambda_\text{phys}\).

---

7 Method comparison

	PINN	Neural ODE	Universal DE
Core idea	t→y mapping; physics = loss	dy/dt = f_θ; solve ODE	dy/dt = g_mech + f_θ; solve ODE
Training	Autograd physics residuals	Backprop through solver	Backprop through solver
Memory	Low (no stored trajectory)	O(N·depth); adjoint → O(depth)	Same as Neural ODE
Extrapolation	Good (physics constrains)	Poor (no physics)	Moderate (partial physics)
Interpretability	θ_mech explicitly estimated	Black box	θ_mech interpretable; correction implicit
Best use case	Known ODE, parameter estimation	Unknown dynamics from rich data	Partially known ODE + unknown forcing

---

8 Extensions

Latent Neural ODEs for irregular time series — (4) extend the Neural ODE framework to an encoder-decoder architecture: a recurrent encoder reads irregularly spaced observations and maps them to a latent initial state; a Neural ODE decoder then generates the latent trajectory forward in time. This is the natural fit for clinical surveillance data with irregular reporting intervals and missing measurements.

Production-quality ODE solvers — The Euler integrator used here is pedagogically transparent but accumulates error over long horizons. The torchdiffeq library (1) provides Runge-Kutta and Dormand-Prince (adaptive step-size) solvers that are fully differentiable and memory-efficient via the adjoint method — a drop-in replacement in the forward pass.

Graph Neural ODEs for spatial networks — Replace the MLP right-hand side with a graph neural network: each node is a geographic unit, edge weights encode mobility flows, and the ODE evolves the entire network state simultaneously. This extends the SIR Neural ODE to metapopulation epidemic models on commuter or air-travel networks.

Real-data application — Dandekar and Barbastathis (5) applied a UDE-like approach to COVID-19 province-level case counts, learning a quarantine-strength function alongside the SIR skeleton. The approach recovered interpretable intervention signals consistent with known lockdown dates — demonstrating that the partial-physics architecture generalises from simulated to noisy real-world data.

Comprehensive reference — Kidger (6) provides a unified treatment of all neural differential equation variants — ODEs, stochastic DEs, and controlled DEs — including the adjoint method, theoretical existence results, and training pathologies. Recommended for readers moving beyond the tutorial level.

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9 References

1.

Chen RTQ, Rubanova Y, Bettencourt J, Duvenaud D. Neural ordinary differential equations. In: Advances in neural information processing systems [Internet]. 2018. Available from: https://arxiv.org/abs/1806.07366

2.

Raissi M, Perdikaris P, Karniadakis GE. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics. 2019;378:686–707. doi:10.1016/j.jcp.2018.10.045

3.

Rackauckas C, Ma Y, Martensen J, Warner C, Zubov K, Supekar R, et al. Universal differential equations for scientific machine learning [Internet]. 2020. Available from: https://arxiv.org/abs/2001.04385

4.

Rubanova Y, Chen RTQ, Duvenaud D. Latent ordinary differential equations for irregularly-sampled time series. In: Advances in neural information processing systems [Internet]. 2019. Available from: https://arxiv.org/abs/1907.03907

5.

Dandekar R, Barbastathis G. Quantifying the effect of quarantine control in COVID-19 infectious spread using machine learning. EBioMedicine. 2020;55:102875. doi:10.1016/j.ebiom.2020.102875

6.

Kidger P. On neural differential equations [Internet]. 2022. Available from: https://arxiv.org/abs/2202.02435

Python     3.12.10
torch      2.11.0+cpu
scipy      1.17.1
matplotlib 3.10.8