2nd/10_Wiki/Topics/AI_and_ML/Physics-informed-Neural-Networks.md

id, title, category, status, canonical_id, aliases, duplicate_of, source_trust_level, confidence_score, verification_status, tags, raw_sources, last_reinforced, github_commit, tech_stack

id	title	category	status	canonical_id	aliases	duplicate_of	source_trust_level	confidence_score	verification_status	tags	raw_sources	last_reinforced	github_commit	tech_stack
wiki-2026-0508-physics-informed-neural-networks	Physics-informed Neural Networks	10_Wiki/Topics	verified	self	PINN physics-informed-NN scientific-ML	none	A	0.9	applied	pinn scientific-ml pde deep-learning		2026-05-10	pending	language framework Python PyTorch / DeepXDE

Physics-informed Neural Networks (PINN)

매 한 줄

"매 NN 매 PDE 의 solution 매 학습 — 매 physics 의 loss 의 embed". 매 Raissi/Perdikaris/Karniadakis (2019, JCP) 매 popularize. 매 soft constraint 매 PDE residual + boundary/initial conditions → 매 mesh-free PDE solver. 매 sparse data + known physics regime 매 best.

매 핵심

매 formulation

• 매 NN: u_θ(x, t) — 매 neural network 매 solution 의 approximate.
• 매 PDE residual loss: L_pde = ||N[u_θ] - 0||² (e.g. ∂u/∂t + u ∂u/∂x - ν ∂²u/∂x² for Burgers).
• 매 BC/IC loss: L_bc + L_ic — 매 boundary + initial conditions.
• 매 data loss (optional): L_data — 매 sparse measurements 의 fit.
• 매 total: L = λ_pde·L_pde + λ_bc·L_bc + λ_ic·L_ic + λ_data·L_data.
• 매 autograd: 매 ∂u/∂x, ∂²u/∂x² 의 torch.autograd.grad 로 compute.

매 강점 + 약점

• 매 강점: mesh-free, 매 inverse problems, 매 sparse/noisy data, 매 high-dim PDEs (curse-resistant), 매 unified forward+inverse.
• 매 약점: 매 stiff PDEs (multi-scale) 매 fail, 매 training slow, 매 loss balancing tricky, 매 FEM 보다 매 large-domain 의 underperform.

매 변형

• 매 XPINN (extended): 매 domain decomposition.
• 매 cPINN (conservative): 매 conservation laws.
• 매 hp-VPINN: variational + hp-refinement.
• 매 PINO (Physics-Informed Neural Operator): 매 operator learning + physics loss.
• 매 DeepONet + PINN hybrid.
• 매 SA-PINN: self-adaptive loss weights.

매 modern (2026)

• 매 Neural Operators (FNO, DeepONet): 매 PINN 보다 매 generalization across PDE families.
• 매 Foundation models for PDEs: Poseidon (2024), DPOT (2024).
• 매 Diffusion-based PDE solvers.
• 매 PINN 매 still useful: 매 inverse problems + 매 physics-constrained design.

매 응용

1. 매 fluid dynamics (Navier-Stokes inverse).
2. 매 cardiovascular flow modeling.
3. 매 subsurface flow (oil/gas).
4. 매 metamaterial inverse design.

💻 패턴

Vanilla PINN — 1D Burgers

import torch, torch.nn as nn

class PINN(nn.Module):
    def __init__(self):
        super().__init__()
        self.net = nn.Sequential(
            nn.Linear(2, 64), nn.Tanh(),
            *[layer for _ in range(4) for layer in (nn.Linear(64, 64), nn.Tanh())],
            nn.Linear(64, 1),
        )
    def forward(self, x, t):
        return self.net(torch.cat([x, t], dim=1))

def pde_residual(model, x, t, nu=0.01/torch.pi):
    x.requires_grad_(True); t.requires_grad_(True)
    u = model(x, t)
    u_t = torch.autograd.grad(u, t, torch.ones_like(u), create_graph=True)[0]
    u_x = torch.autograd.grad(u, x, torch.ones_like(u), create_graph=True)[0]
    u_xx = torch.autograd.grad(u_x, x, torch.ones_like(u_x), create_graph=True)[0]
    return u_t + u * u_x - nu * u_xx

Loss assembly + training

model = PINN().cuda()
opt = torch.optim.Adam(model.parameters(), lr=1e-3)

x_f = torch.rand(10000, 1, device="cuda") * 2 - 1   # collocation points
t_f = torch.rand(10000, 1, device="cuda")
x_b = torch.tensor([[-1.], [1.]], device="cuda")     # boundary
t_b = torch.rand(2, 1, device="cuda")
x_i = torch.linspace(-1, 1, 200, device="cuda").unsqueeze(1)  # initial t=0
t_i = torch.zeros_like(x_i)

for it in range(20000):
    opt.zero_grad()
    L_pde = (pde_residual(model, x_f, t_f)**2).mean()
    L_bc  = (model(x_b, t_b)**2).mean()
    L_ic  = ((model(x_i, t_i) + torch.sin(torch.pi * x_i))**2).mean()
    L = L_pde + 10*L_bc + 10*L_ic
    L.backward(); opt.step()

DeepXDE — 매 high-level library

import deepxde as dde, numpy as np

def pde(x, u):
    du_x = dde.grad.jacobian(u, x, i=0, j=0)
    du_t = dde.grad.jacobian(u, x, i=0, j=1)
    du_xx = dde.grad.hessian(u, x, i=0, j=0)
    return du_t + u * du_x - 0.01/np.pi * du_xx

geom = dde.geometry.Interval(-1, 1)
timedomain = dde.geometry.TimeDomain(0, 1)
geomtime = dde.geometry.GeometryXTime(geom, timedomain)

bc = dde.icbc.DirichletBC(geomtime, lambda x: 0, lambda _, on_b: on_b)
ic = dde.icbc.IC(geomtime, lambda x: -np.sin(np.pi*x[:,0:1]), lambda _, on_i: on_i)

data = dde.data.TimePDE(geomtime, pde, [bc, ic], num_domain=2540, num_boundary=80, num_initial=160)
net = dde.nn.FNN([2] + [64]*4 + [1], "tanh", "Glorot normal")
model = dde.Model(data, net)
model.compile("adam", lr=1e-3); model.train(iterations=15000)
model.compile("L-BFGS"); model.train()

Inverse problem — discover ν

# Treat viscosity as trainable parameter
nu = torch.nn.Parameter(torch.tensor(0.05, device="cuda"))
opt = torch.optim.Adam([*model.parameters(), nu], lr=1e-3)

# Add data loss from sparse measurements
x_d, t_d, u_d = load_measurements()
L_data = ((model(x_d, t_d) - u_d)**2).mean()
# nu converges to true value during training

Self-adaptive weights (SA-PINN)

# Trainable per-point weights — 매 hard regions 의 emphasize
lambda_pde = nn.Parameter(torch.ones(N_collocation, 1, device="cuda"))
L_pde = (lambda_pde * pde_residual(...)**2).mean()
# Maximize w.r.t. lambda, minimize w.r.t. theta — adversarial-ish

Fourier features — 매 high-frequency PDE

class FourierEmbed(nn.Module):
    def __init__(self, in_dim, mapping_size=64, scale=10):
        super().__init__()
        self.B = nn.Parameter(torch.randn(in_dim, mapping_size) * scale, requires_grad=False)
    def forward(self, x):
        proj = 2 * torch.pi * x @ self.B
        return torch.cat([torch.sin(proj), torch.cos(proj)], dim=-1)

매 결정 기준

상황	방법
매 known PDE + sparse data	매 PINN
매 inverse problem (parameter ID)	매 PINN (best fit)
매 solve same PDE many times	매 FNO / DeepONet
매 large-scale forward solve	매 FEM/FVM (still better)
매 stiff multi-scale	매 PINN 매 careful (XPINN, sequence-to-sequence)
매 high-D Black-Scholes	매 PINN ≫ FEM

기본값: 매 DeepXDE (forward+inverse, well-engineered).

🔗 Graph

🤖 LLM 활용

언제: 매 forward/inverse PDE problems 매 sparse data, 매 mesh-free desired, 매 unified data+physics framework 의 wanted. 언제 X: 매 production CFD (FEM/FVM 매 mature), 매 stiff multi-scale (notorious failure), 매 same PDE solved 1000s of times (operator learning).

❌ 안티패턴

• 매 unbalanced loss weights: 매 L_pde dominates → 매 BC violation. 매 SA-PINN 또는 manual tuning.
• 매 single-stage stiff training: 매 sequence-to-sequence (causal) training 의 사용.
• 매 ReLU activation: 매 PINN 매 smooth activation (tanh, sin, GELU) 의 필요 — autograd hessian.
• 매 PINN > FEM claim: 매 standard forward problems 매 still FEM-superior.

🧪 검증 / 중복

• Verified (Raissi 2019 JCP, DeepXDE docs, NVIDIA Modulus).
• 신뢰도 A.

🕓 Changelog

날짜	변경
2026-05-08	Phase 1
2026-05-10	Manual cleanup — PINN formulation + DeepXDE + inverse + SA-PINN