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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8"/>
<meta name="viewport" content="width=device-width, initial-scale=1.0"/>
<title>PINNeAPPle — Learning Course</title>
<style>
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--text:#e6edf3; --muted:#8b949e; --tier1:#58a6ff; --tier2:#ffa657; --tier3:#3fb950; --nemo:#d2a8ff;
}
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/* header pin */
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/* Tier color blocks */
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/* Lesson card */
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.lesson-body p{font-size:12px;color:var(--muted);}
/* Result chart mock */
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/* Flow / steps */
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.step-content strong{font-size:13px;color:var(--text);}
.step-content p{font-size:12px;color:var(--muted);margin-top:2px;}
/* Comparison table accent */
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.worse{color:var(--accent3);}
.same{color:var(--muted);}
/* Nav */
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/* result mockup */
.result-plot{background:linear-gradient(135deg,var(--surface) 0%,var(--surface2) 100%);border:1px solid var(--border);border-radius:10px;padding:14px;display:flex;flex-direction:column;gap:8px;overflow:hidden;}
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.sparkline{height:60px;position:relative;}
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.section-intro p{font-size:13px;color:var(--muted);}
</style>
</head>
<body>
<div id="progress-bar"></div>
<div class="deck">
<!-- ========== S1: TITLE ========== -->
<div class="slide active" id="s1">
<div class="hero-bg"></div>
<div class="slide-header">
<span class="slide-tag">PINNeAPPle Learning</span>
<div class="divider-line"></div>
<span class="slide-num">1 / 20</span>
</div>
<div class="hero-center">
<div style="font-size:12px;color:var(--accent);letter-spacing:3px;text-transform:uppercase;font-weight:700;margin-bottom:14px;">Physics AI — Step by Step</div>
<div class="hero-title">Physics AI Course</div>
<div style="font-size:28px;font-weight:800;color:var(--accent4);letter-spacing:-0.8px;margin-top:4px;">with PINNeAPPle</div>
<div class="hero-sub">12 lessons · From your first PINN to NVIDIA PhysicsNeMo<br>Hands-on code · Real physics · Production-ready workflow</div>
<div class="badge-row" style="justify-content:center;margin-top:16px;">
<span class="badge t1" style="color:var(--tier1)">Tier 1 — Explorer</span>
<span class="badge orange">Tier 2 — Experimenter</span>
<span class="badge green">Tier 3 — Builder</span>
<span class="badge purple">→ PhysicsNeMo</span>
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<div class="stat-grid" style="margin-top:28px;max-width:620px;">
<div class="stat-box"><div class="num">12</div><div class="label">Lessons</div></div>
<div class="stat-box"><div class="num">8</div><div class="label">PDE Problems</div></div>
<div class="stat-box"><div class="num" style="font-size:28px;">10+</div><div class="label">PINNeAPPle APIs</div></div>
<div class="stat-box"><div class="num">1</div><div class="label">Destination</div></div>
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</div>
<div style="text-align:center;font-size:12px;color:var(--muted);position:relative;z-index:1;">April 2026 · Yan Barros · <code style="color:var(--accent);">python -m pinneaple_learning.course.lesson_01_first_pinn</code></div>
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<!-- ========== S2: LEARNING PATH ========== -->
<div class="slide" id="s2">
<div class="slide-header"><span class="slide-tag">Overview</span><div class="divider-line"></div><span class="slide-num">2 / 20</span></div>
<h2>The Learning Path</h2>
<div class="subtitle" style="margin-bottom:14px;">Three progressive tiers — each one builds on the previous. PhysicsNeMo is the destination.</div>
<div class="body">
<div class="four-col">
<div class="tier-block t1">
<h4>Tier 1 — Explorer</h4>
<ul class="bullets" style="font-size:12px;">
<li>Lesson 01 — First PINN</li>
<li>Lesson 02 — Loss functions</li>
<li>Lesson 03 — Heat equation</li>
<li>Lesson 04 — Geometry CSG</li>
<li>Lesson 05 — 2D Poisson</li>
</ul>
<div class="badge-row" style="margin-top:10px;"><span class="badge" style="color:var(--tier1);border-color:rgba(88,166,255,.3);">SymbolicPDE · HardBC</span></div>
</div>
<div class="tier-block t2">
<h4>Tier 2 — Experimenter</h4>
<ul class="bullets" style="font-size:12px;">
<li>Lesson 06 — Architectures</li>
<li>Lesson 07 — Inverse problem</li>
<li>Lesson 08 — UQ</li>
<li>Lesson 09 — Time marching</li>
</ul>
<div class="badge-row" style="margin-top:10px;"><span class="badge orange">SIREN · EKI · Ensemble</span></div>
</div>
<div class="tier-block t3">
<h4>Tier 3 — Builder</h4>
<ul class="bullets" style="font-size:12px;">
<li>Lesson 10 — Validation</li>
<li>Lesson 11 — Operators</li>
<li>Lesson 12 — Production</li>
</ul>
<div class="badge-row" style="margin-top:10px;"><span class="badge green">PhysicsValidator · ONNX</span></div>
</div>
<div class="tier-block nemo" style="border-color:rgba(210,168,255,.35);">
<h4>Advanced — Co-Simulation</h4>
<ul class="bullets" style="font-size:12px;">
<li>Lesson 13 — Co-sim graphs</li>
<li>Lesson 14 — Multi-physics BPTT</li>
<li>pinneaple_cosim engine</li>
<li>→ PhysicsNeMo / scale</li>
</ul>
<div class="badge-row" style="margin-top:10px;"><span class="badge purple">NEW · pinneaple_cosim</span></div>
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</div>
<div class="hcard">
<h4>How to run the course</h4>
<pre style="margin-top:4px;"><span class="kw">import</span> pinneaple_learning.course <span class="kw">as</span> course
course.<span class="fn">list_lessons</span>() <span class="cm"># show all 12 lessons</span>
course.<span class="fn">run_lesson</span>(<span class="nu">1</span>) <span class="cm"># run lesson 1 interactively</span>
<span class="cm"># Or from terminal:</span>
<span class="cm"># python -m pinneaple_learning.course.lesson_01_first_pinn</span></pre>
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<!-- ========== S3: LESSON 01 ========== -->
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<div class="slide-header"><span class="slide-tag t1">Lesson 01</span><div class="divider-line"></div><span class="slide-num">3 / 20</span></div>
<h2>Your First PINN — Harmonic Oscillator</h2>
<div class="subtitle" style="margin-bottom:12px;">x″(t) + ω²x(t) = 0 · x(0)=1, x′(0)=0 · Exact: cos(ωt)</div>
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<h4>Key PINNeAPPle APIs</h4>
<pre><span class="kw">from</span> pinneaple_symbolic <span class="kw">import</span> <span class="cl">SymbolicPDE</span>, <span class="cl">HardBC</span>
<span class="kw">import</span> sympy <span class="kw">as</span> sp
<span class="cm"># Step 1 — Define PDE symbolically</span>
t, x = sp.<span class="fn">Symbol</span>(<span class="st">'t'</span>), sp.<span class="fn">Function</span>(<span class="st">'x'</span>)
pde = <span class="cl">SymbolicPDE</span>(
expr = x(t).diff(t,<span class="nu">2</span>) + omega**<span class="nu">2</span> * x(t),
coord_syms = [t], field_syms = [x],
)
residual_fn = pde.<span class="fn">to_residual_fn</span>(net)
<span class="cm"># Step 2 — Hard-enforce IC: x(0)=1 via ansatz</span>
<span class="cm"># x_wrapped(t) = 1 + t·net(t) → always 1 at t=0</span>
hard_ic = <span class="cl">HardBC</span>(
distance_fn = <span class="kw">lambda</span> t: t,
bc_value_fn = <span class="kw">lambda</span> t: torch.<span class="fn">ones_like</span>(t),
)
net_wrapped = hard_ic.<span class="fn">wrap_model</span>(net)
<span class="cm"># Step 3 — Training loss (position IC enforced for free)</span>
r = residual_fn(t_col) <span class="cm"># PDE residual</span>
loss_vel = (xd_at_0 - <span class="nu">0</span>).<span class="fn">pow</span>(<span class="nu">2</span>) <span class="cm"># velocity IC only</span>
loss = r.<span class="fn">pow</span>(<span class="nu">2</span>).<span class="fn">mean</span>() + <span class="nu">200</span> * loss_vel</pre>
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<div class="card">
<h4>What SymbolicPDE does for you</h4>
<ul class="bullets" style="font-size:12px;">
<li>Write the PDE once in SymPy — no manual autograd chains</li>
<li><code style="color:var(--accent4)">to_residual_fn(net)</code> returns a closure that evaluates F[u_θ]</li>
<li>Works for any ODE/PDE expressible in SymPy</li>
<li>HardBC removes L_pos entirely — one fewer hyperparameter</li>
</ul>
</div>
</div>
<div>
<div class="result-plot" style="margin-bottom:12px;">
<div class="plot-title">Lesson 01 result — ω=2, T=4π</div>
<!-- SVG mock of cos(ωt) vs PINN prediction -->
<svg viewBox="0 0 300 120" style="width:100%;height:130px;">
<defs><linearGradient id="g1" x1="0" y1="0" x2="1" y2="0"><stop offset="0%" stop-color="#58a6ff"/><stop offset="100%" stop-color="#d2a8ff"/></linearGradient></defs>
<!-- axes -->
<line x1="20" y1="100" x2="290" y2="100" stroke="#30363d" stroke-width="1"/>
<line x1="20" y1="10" x2="20" y2="100" stroke="#30363d" stroke-width="1"/>
<!-- exact cos(2t) — 2 full periods over 0..4π≈12.57 -->
<path d="M20,55 C40,15 60,95 80,55 C100,15 120,95 140,55 C160,15 180,95 200,55 C220,15 240,95 260,55 C280,15 290,55 290,55" stroke="#e6edf3" stroke-width="2" fill="none"/>
<!-- PINN dashed -->
<path d="M20,55 C41,17 61,93 81,56 C101,16 121,94 141,56 C161,16 181,94 201,57 C221,17 241,93 261,56 C281,16 290,56 290,56" stroke="url(#g1)" stroke-width="1.5" fill="none" stroke-dasharray="5,3"/>
<text x="22" y="115" fill="#8b949e" font-size="9">0</text>
<text x="270" y="115" fill="#8b949e" font-size="9">4π</text>
<text x="22" y="19" fill="#e6edf3" font-size="9">exact</text>
<text x="60" y="19" fill="url(#g1)" font-size="9">PINN</text>
</svg>
</div>
<div class="metric-row">
<div class="metric"><div class="val" style="color:var(--accent2);">1.2e-3</div><div class="lbl">Relative L2</div></div>
<div class="metric"><div class="val" style="color:var(--accent);">8k</div><div class="lbl">Epochs</div></div>
<div class="metric"><div class="val" style="color:var(--accent4);">0</div><div class="lbl">IC Error (Hard)</div></div>
<div class="metric"><div class="val" style="color:var(--accent5);">2</div><div class="lbl">Loss Terms</div></div>
</div>
<div class="card" style="margin-top:10px;">
<h4>Manual vs PINNeAPPle</h4>
<table>
<tr><th>Aspect</th><th>Manual PyTorch</th><th>PINNeAPPle</th></tr>
<tr><td>PDE residual</td><td class="same">Manual autograd</td><td class="better">SymbolicPDE</td></tr>
<tr><td>IC x(0)=1</td><td class="worse">Penalty term λ·(u0-1)²</td><td class="better">HardBC — exact</td></tr>
<tr><td>Loss terms</td><td class="same">3 terms</td><td class="better">2 terms</td></tr>
<tr><td>Hyperparams</td><td class="worse">λ_pos + λ_vel</td><td class="better">λ_vel only</td></tr>
</table>
</div>
</div>
</div>
</div>
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<!-- ========== S4: LESSON 02-03 ========== -->
<div class="slide" id="s4">
<div class="slide-header"><span class="slide-tag t1">Lessons 02–03</span><div class="divider-line"></div><span class="slide-num">4 / 20</span></div>
<h2>Loss Anatomy + Heat Equation Forward Solve</h2>
<div class="subtitle" style="margin-bottom:12px;">Understanding what each term does · TwoPhaseTrainer · compare_to_analytical</div>
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<div class="hcard" style="margin-bottom:12px;">
<h4>Lesson 02 — Loss Function Anatomy</h4>
<pre><span class="kw">from</span> pinneaple_symbolic <span class="kw">import</span> <span class="cl">DirichletBC</span>
<span class="kw">from</span> pinneaple_train <span class="kw">import</span> <span class="cl">SelfAdaptiveWeights</span>
<span class="cm"># Manual weights — needs tuning</span>
loss = l_pde + <span class="nu">10.0</span>*l_bc + <span class="nu">100.0</span>*l_ic
<span class="cm"># vs SelfAdaptiveWeights — auto-balancing</span>
saw = <span class="cl">SelfAdaptiveWeights</span>(
loss_names=[<span class="st">"pde"</span>, <span class="st">"bc"</span>, <span class="st">"ic"</span>], alpha=<span class="nu">0.9</span>
)
loss = saw({<span class="st">"pde"</span>: l_pde, <span class="st">"bc"</span>: l_bc, <span class="st">"ic"</span>: l_ic})
<span class="cm"># DirichletBC.loss() handles the BC term</span>
bc = <span class="cl">DirichletBC</span>(value_fn=<span class="kw">lambda</span> xt: zeros)
l_bc = bc.<span class="fn">loss</span>(net, x_boundary)</pre>
</div>
<div class="card">
<h4>Lesson 03 — TwoPhaseTrainer</h4>
<pre style="font-size:11px;"><span class="kw">from</span> pinneaple_train <span class="kw">import</span> <span class="cl">TwoPhaseTrainer</span>, <span class="cl">TwoPhaseConfig</span>
cfg = <span class="cl">TwoPhaseConfig</span>(
phase1_epochs=<span class="nu">5_000</span>, <span class="cm"># Adam warm-up</span>
phase2_epochs=<span class="nu">1_000</span>, <span class="cm"># L-BFGS refinement</span>
phase1_lr=<span class="nu">1e-3</span>, phase2_lr=<span class="nu">1.0</span>,
)
trainer = <span class="cl">TwoPhaseTrainer</span>(net, pinn_loss, cfg)
history = trainer.<span class="fn">train</span>()
<span class="cm"># Validate against exact solution</span>
<span class="kw">from</span> pinneaple_validate <span class="kw">import</span> compare_to_analytical
m = <span class="fn">compare_to_analytical</span>(
model=net, analytical_fn=exact,
coord_names=[<span class="st">"x"</span>,<span class="st">"t"</span>],
domain_bounds={<span class="st">"x"</span>:(<span class="nu">0</span>,<span class="nu">1</span>),<span class="st">"t"</span>:(<span class="nu">0</span>,<span class="nu">1</span>)},
)</pre>
</div>
</div>
<div>
<div class="result-plot" style="margin-bottom:12px;">
<div class="plot-title">Lesson 02 — L2 error by weight strategy</div>
<div class="bar-chart" style="margin-top:24px;">
<div class="bar-wrap"><div class="bar" style="height:85px;background:#d62728;" data-val="8.3e-2"></div><div class="bar-label">λ=1</div></div>
<div class="bar-wrap"><div class="bar" style="height:28px;background:#2ca02c;" data-val="7.1e-3"></div><div class="bar-label">λ=100</div></div>
<div class="bar-wrap"><div class="bar" style="height:42px;background:#ff7f0e;" data-val="1.9e-2"></div><div class="bar-label">λ=1000</div></div>
<div class="bar-wrap"><div class="bar" style="height:18px;background:var(--accent);" data-val="4.2e-3"></div><div class="bar-label">SAW</div></div>
</div>
</div>
<div class="result-plot">
<div class="plot-title">Lesson 03 — Heat equation · α=0.05</div>
<svg viewBox="0 0 300 100" style="width:100%;height:100px;">
<!-- Heat solution u(x,t)=exp(-π²αt)sin(πx): always positive, decays -->
<!-- x-axis at y=92; profiles are symmetric arches above baseline -->
<line x1="20" y1="92" x2="290" y2="92" stroke="#30363d" stroke-width="1"/>
<!-- t=0.0 : exp(0)=1.0 → peak at y=6 -->
<path d="M20,92 Q155,6 290,92" stroke="#e6edf3" stroke-width="2" fill="none" opacity="0.95"/>
<!-- t=0.2 : exp(-π²·0.05·0.2)≈0.91 → peak at y=14 -->
<path d="M20,92 Q155,14 290,92" stroke="#3fb950" stroke-width="1.5" fill="none" opacity="0.85"/>
<!-- t=0.5 : exp(-0.247)≈0.78 → peak at y=26 -->
<path d="M20,92 Q155,26 290,92" stroke="#58a6ff" stroke-width="1.5" fill="none" opacity="0.75"/>
<!-- t=0.8 : exp(-0.395)≈0.67 → peak at y=37 -->
<path d="M20,92 Q155,37 290,92" stroke="#ffa657" stroke-width="1.5" fill="none" opacity="0.65"/>
<!-- t=1.0 : exp(-0.493)≈0.61 → peak at y=44 -->
<path d="M20,92 Q155,44 290,92" stroke="#d2a8ff" stroke-width="1.5" fill="none" opacity="0.55"/>
<text x="22" y="10" fill="#8b949e" font-size="9">t=0 → t=1 (α=0.05, decaying amplitude)</text>
<text x="215" y="10" fill="#3fb950" font-size="9">L2 = 6.1e-4</text>
<text x="22" y="98" fill="#8b949e" font-size="8">x=0</text>
<text x="272" y="98" fill="#8b949e" font-size="8">x=1</text>
</svg>
<div class="metric-row" style="margin-top:6px;">
<div class="metric"><div class="val" style="color:var(--accent2);font-size:20px;">6.1e-4</div><div class="lbl">Rel L2</div></div>
<div class="metric"><div class="val" style="color:var(--accent);font-size:20px;">6k</div><div class="lbl">Epochs</div></div>
<div class="metric"><div class="val" style="color:var(--accent4);font-size:20px;">Adam+LBFGS</div><div class="lbl">Trainer</div></div>
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<!-- ========== S5: LESSON 04-05 ========== -->
<div class="slide" id="s5">
<div class="slide-header"><span class="slide-tag t1">Lessons 04–05</span><div class="divider-line"></div><span class="slide-num">5 / 20</span></div>
<h2>Geometry Generation + 2D Poisson on L-shape</h2>
<div class="subtitle" style="margin-bottom:12px;">CSG domains · SDF-based HardBC · −∇²u = 1</div>
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<div class="two-col">
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<div class="hcard" style="margin-bottom:12px;">
<h4>Lesson 04 — CSG with PINNeAPPle</h4>
<pre><span class="kw">from</span> pinneaple_geom <span class="kw">import</span> (
<span class="cl">CSGRectangle</span>, <span class="cl">CSGCircle</span>,
<span class="cl">lshape</span>, <span class="cl">channel_with_hole</span>
)
<span class="cm"># Boolean operations — Python operators</span>
domain = <span class="cl">CSGRectangle</span>(<span class="nu">0</span>,<span class="nu">0</span>,<span class="nu">2</span>,<span class="nu">1</span>) - <span class="cl">CSGCircle</span>(<span class="nu">1</span>,<span class="nu">.5</span>,<span class="nu">.3</span>)
<span class="cm"># Sampling: rejection-sample from SDF</span>
pts_col = domain.<span class="fn">sample_interior</span>(<span class="nu">2000</span>, seed=<span class="nu">42</span>)
pts_bnd = domain.<span class="fn">sample_boundary</span>(<span class="nu">500</span>, seed=<span class="nu">42</span>)
<span class="cm"># SDF for hard BC enforcement</span>
sdf_vals = domain.<span class="fn">sdf</span>(pts_col) <span class="cm"># −inside, +outside</span>
<span class="cm"># Pre-built complex domains</span>
ch = <span class="fn">channel_with_hole</span>() <span class="cm"># flow past cylinder</span>
ls = <span class="fn">lshape</span>() <span class="cm"># re-entrant corner</span></pre>
</div>
<div class="card">
<h4>Lesson 05 — SDF-based HardBC</h4>
<pre style="font-size:11px;"><span class="cm"># u(x) = SDF(x)·net(x) → u=0 on boundary EXACTLY</span>
hard_bc = <span class="cl">HardBC</span>(
distance_fn = <span class="kw">lambda</span> xy: -domain.<span class="fn">sdf</span>(xy).clamp(min=<span class="nu">0</span>),
bc_value_fn = <span class="kw">lambda</span> xy: zeros,
)
net = hard_bc.<span class="fn">wrap_model</span>(net_inner)
<span class="cm"># No L_bc term needed — only L_pde remains:</span>
def loss_fn(model):
xy_col = domain.<span class="fn">sample_interior</span>(<span class="nu">4000</span>)
r = poisson_residual(model, xy_col)
<span class="kw">return</span> r.<span class="fn">pow</span>(<span class="nu">2</span>).<span class="fn">mean</span>() <span class="cm"># ONE term only</span></pre>
</div>
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<div>
<div style="display:grid;grid-template-columns:1fr 1fr;gap:10px;margin-bottom:12px;">
<!-- Domain grid mockup -->
<div class="chart-area">
<div class="chart-area" style="border:none;padding:0;">
<div class="plot-title">CSG Domains (Lesson 04)</div>
<svg viewBox="0 0 120 130" style="width:100%;height:130px;">
<!-- Rect -->
<rect x="5" y="5" width="50" height="30" fill="none" stroke="#58a6ff" stroke-width="1.5"/>
<text x="8" y="17" fill="#58a6ff" font-size="6">Rectangle</text>
<circle cx="30" cy="42" r="10" fill="none" stroke="#ffa657" stroke-width="1.5"/>
<text x="22" y="59" fill="#ffa657" font-size="6">Circle</text>
<!-- Rect - Circle (L-shape like) -->
<rect x="65" y="5" width="50" height="30" fill="rgba(88,166,255,.1)" stroke="#58a6ff" stroke-width="1"/>
<circle cx="90" cy="20" r="8" fill="#0d1117" stroke="#30363d" stroke-width="1"/>
<text x="67" y="44" fill="#3fb950" font-size="6">Rect − Circle</text>
<!-- L-shape -->
<polygon points="5,70 55,70 55,85 30,85 30,110 5,110" fill="rgba(255,166,87,.08)" stroke="#ffa657" stroke-width="1.5"/>
<text x="7" y="120" fill="#ffa657" font-size="6">L-Shape</text>
<!-- Channel with hole -->
<rect x="65" y="70" width="50" height="25" fill="rgba(210,168,255,.08)" stroke="#d2a8ff" stroke-width="1"/>
<circle cx="90" cy="82" r="5" fill="#0d1117" stroke="#30363d" stroke-width="1"/>
<text x="67" y="107" fill="#d2a8ff" font-size="6">Channel+hole</text>
</svg>
</div>
</div>
<div class="chart-area">
<div class="plot-title">L-shape Poisson |Error|</div>
<svg viewBox="0 0 120 130" style="width:100%;height:130px;">
<!-- L-shape filled with "solution" -->
<polygon points="5,10 85,10 85,55 50,55 50,120 5,120" fill="rgba(88,166,255,.05)" stroke="#30363d" stroke-width="1"/>
<!-- Error heatmap mockup — corner singularity -->
<radialGradient id="corner" cx="85%" cy="43%" r="40%">
<stop offset="0%" stop-color="#f78166" stop-opacity=".8"/>
<stop offset="50%" stop-color="#ffa657" stop-opacity=".4"/>
<stop offset="100%" stop-color="transparent"/>
</radialGradient>
<polygon points="5,10 85,10 85,55 50,55 50,120 5,120" fill="url(#corner)"/>
<text x="7" y="130" fill="#8b949e" font-size="7">Re-entrant corner</text>
<text x="60" y="30" fill="#f78166" font-size="7">high err</text>
<text x="10" y="80" fill="#3fb950" font-size="7">low err</text>
</svg>
</div>
</div>
<div class="metric-row">
<div class="metric"><div class="val" style="color:var(--accent2);font-size:18px;">0</div><div class="lbl">BC residual (HardBC)</div></div>
<div class="metric"><div class="val" style="color:var(--accent);font-size:18px;">2.4e-3</div><div class="lbl">BC (Penalty)</div></div>
<div class="metric"><div class="val" style="color:var(--accent4);font-size:18px;">8</div><div class="lbl">Domains built</div></div>
<div class="metric"><div class="val" style="color:var(--accent5);font-size:18px;">1</div><div class="lbl">Loss term (HardBC)</div></div>
</div>
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<!-- ========== S6: LESSON 06 ARCHITECTURES ========== -->
<div class="slide" id="s6">
<div class="slide-header"><span class="slide-tag t2">Lesson 06</span><div class="divider-line"></div><span class="slide-num">6 / 20</span></div>
<h2>Model Architectures — SIREN vs ModifiedMLP vs Fourier</h2>
<div class="subtitle" style="margin-bottom:12px;">Helmholtz u″ + k²u = 0 · k=8 · 8 full oscillations in [0, 2π/k]</div>
<div class="body">
<div class="two-col">
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<div class="hcard" style="margin-bottom:12px;">
<h4>PINNeAPPle model imports</h4>
<pre><span class="kw">from</span> pinneaple_models <span class="kw">import</span> (
<span class="cl">SIREN</span>, <span class="cl">ModifiedMLP</span>, <span class="cl">FourierFeatureEmbedding</span>
)
<span class="cm"># SIREN — periodic activations sin(ω₀·Wx + b)</span>
siren = <span class="cl">SIREN</span>(in_dim=<span class="nu">1</span>, out_dim=<span class="nu">1</span>,
hidden_dim=<span class="nu">128</span>, n_layers=<span class="nu">5</span>,
omega_0=<span class="nu">30.0</span>) <span class="cm"># ω₀=30 standard init</span>
<span class="cm"># ModifiedMLP — Fourier embed + highway gating</span>
modmlp = <span class="cl">ModifiedMLP</span>(in_dim=<span class="nu">1</span>, out_dim=<span class="nu">1</span>,
hidden_dim=<span class="nu">128</span>, n_fourier=<span class="nu">32</span>,
sigma=<span class="nu">12.0</span>) <span class="cm"># σ ≈ k×1.5</span>
<span class="cm"># FourierFeatureEmbedding — prepend to any MLP</span>
embed = <span class="cl">FourierFeatureEmbedding</span>(in_dim=<span class="nu">1</span>,
n_fourier=<span class="nu">64</span>,
sigma=<span class="nu">12.0</span>)
<span class="cm"># Maps x → [cos(2πBx), sin(2πBx)] (128-dim)</span>
<span class="cm"># NOTE: SIREN / ModifiedMLP return ModelOutput</span>
u = siren(x_col).<span class="at">y</span> <span class="cm"># extract .y to get tensor</span></pre>
</div>
<div class="card">
<h4>When to use what</h4>
<table style="font-size:12px;">
<tr><th>Architecture</th><th>Best for</th><th>Cost</th></tr>
<tr><td>Tanh MLP</td><td>k < 4, smooth PDEs</td><td class="better">Lowest</td></tr>
<tr><td>SIREN</td><td>High-freq, wave, Helmholtz</td><td class="same">Medium</td></tr>
<tr><td>ModifiedMLP</td><td>Multi-scale, long domain</td><td class="same">Medium</td></tr>
<tr><td>Fourier+MLP</td><td>Quick fix for spectral bias</td><td class="better">Low</td></tr>
</table>
</div>
</div>
<div>
<div class="result-plot" style="margin-bottom:12px;">
<div class="plot-title">Helmholtz k=8 — Architecture benchmark</div>
<svg viewBox="0 0 280 130" style="width:100%;height:140px;">
<!-- 4 prediction curves vs exact sin(8x) -->
<!-- exact sin(8x) — many oscillations in [0, 0.785] -->
<path d="M15,65 C28,30 40,100 55,65 C68,30 82,100 95,65 C108,30 122,100 135,65 C148,30 162,100 175,65 C188,30 202,100 215,65 C228,30 242,100 260,65" stroke="#e6edf3" stroke-width="2.5" fill="none" opacity=".9"/>
<!-- Tanh MLP — fails (flat or wrong) -->
<path d="M15,65 Q60,62 100,66 Q140,64 180,66 Q220,64 260,65" stroke="#d62728" stroke-width="1.5" fill="none" stroke-dasharray="4,2"/>
<!-- SIREN — good match -->
<path d="M15,65 C28,32 40,98 55,66 C68,32 82,98 95,66 C108,32 122,98 135,66 C148,32 162,98 175,66 C188,32 202,98 215,66 C228,32 242,98 261,66" stroke="#3fb950" stroke-width="1.5" fill="none" stroke-dasharray="6,2"/>
<!-- ModifiedMLP — good -->
<path d="M15,65 C28,33 40,97 55,66 C68,33 82,97 95,66 C108,33 122,97 135,66 C148,33 162,97 175,67 C188,33 202,97 215,66 C228,33 242,97 260,65" stroke="#58a6ff" stroke-width="1.5" fill="none" stroke-dasharray="8,2"/>
<!-- Fourier+MLP -->
<path d="M15,65 C28,35 40,95 55,66 C68,35 82,95 95,66 C108,35 122,95 135,66 C148,35 162,95 175,67 C188,35 202,95 215,66 C228,35 242,95 260,66" stroke="#ffa657" stroke-width="1.5" fill="none" stroke-dasharray="3,2"/>
<line x1="15" y1="120" x2="265" y2="120" stroke="#30363d" stroke-width="1"/>
<!-- Legend -->
<text x="15" y="12" fill="#e6edf3" font-size="8">Exact</text>
<text x="50" y="12" fill="#d62728" font-size="8">Tanh MLP</text>
<text x="105" y="12" fill="#3fb950" font-size="8">SIREN</text>
<text x="148" y="12" fill="#58a6ff" font-size="8">ModMLP</text>
<text x="193" y="12" fill="#ffa657" font-size="8">Fourier</text>
</svg>
</div>
<div class="card">
<h4>Benchmark — Helmholtz k=8, 10k epochs</h4>
<table style="font-size:12px;">
<tr><th>Architecture</th><th>L2 error</th><th>Params</th></tr>
<tr><td>Tanh MLP (3×128)</td><td class="worse">4.2e-1</td><td>33k</td></tr>
<tr><td>SIREN (5×128)</td><td class="better">3.8e-3</td><td>50k</td></tr>
<tr><td>ModifiedMLP (6×128)</td><td class="better">5.1e-3</td><td>74k</td></tr>
<tr><td>Fourier+MLP (3×128)</td><td class="pp">1.2e-2</td><td>50k</td></tr>
</table>
</div>
</div>
</div>
</div>
</div>
<!-- ========== S7: LESSON 07 INVERSE ========== -->
<div class="slide" id="s7">
<div class="slide-header"><span class="slide-tag t2">Lesson 07</span><div class="divider-line"></div><span class="slide-num">7 / 20</span></div>
<h2>Inverse Problem — Recovering Diffusivity</h2>
<div class="subtitle" style="margin-bottom:12px;">u_t = α u_xx · unknown α · 50 noisy observations · two recovery methods</div>
<div class="body">
<div class="two-col">
<div>
<div class="hcard" style="margin-bottom:12px;">
<h4>Approach 1 — Gradient-based (α as nn.Parameter)</h4>
<pre><span class="cm"># α is a learnable parameter — same as network weights</span>
alpha = nn.<span class="cl">Parameter</span>(torch.<span class="fn">tensor</span>([<span class="nu">0.2</span>])) <span class="cm"># wrong init</span>
opt = Adam([*net.parameters(), alpha], lr=<span class="nu">1e-3</span>)
loss = l_pde(alpha) + <span class="nu">100</span>*l_ic + <span class="nu">1000</span>*l_data
<span class="cm"># Gradient flows through BOTH net weights AND alpha</span>
alpha.data.<span class="fn">clamp_</span>(<span class="nu">1e-4</span>, <span class="nu">1.0</span>) <span class="cm"># keep physical</span></pre>
</div>
<div class="hcard">
<h4>Approach 2 — EKI (derivative-free)</h4>
<pre><span class="kw">from</span> pinneaple_inverse <span class="kw">import</span> (
<span class="cl">EnsembleKalmanInversion</span>, <span class="cl">EKIConfig</span>
)
<span class="cm"># Only need: forward_fn(theta) → u predictions</span>
eki_cfg = <span class="cl">EKIConfig</span>(
n_ensemble=<span class="nu">20</span>, n_iterations=<span class="nu">15</span>,
noise_std=<span class="nu">0.01</span>, init_spread=<span class="nu">0.5</span>,
)
eki = <span class="cl">EnsembleKalmanInversion</span>(
forward_fn=forward_model, config=eki_cfg
)
history = eki.<span class="fn">solve</span>(y_obs=u_obs, theta_init=theta0)
alpha_eki = exp(history.theta_mean_history[-<span class="nu">1</span>])</pre>
</div>
</div>
<div>
<div class="result-plot" style="margin-bottom:12px;">
<div class="plot-title">α convergence — gradient-based (5000 epochs)</div>
<svg viewBox="0 0 280 100" style="width:100%;height:100px;">
<!-- α starts at 0.2, converges to 0.05 -->
<path d="M15,20 C40,20 60,25 90,55 C120,72 150,78 200,80 C230,81 260,81 275,81" stroke="#58a6ff" stroke-width="2" fill="none"/>
<line x1="15" y1="81" x2="275" y2="81" stroke="#3fb950" stroke-width="1.5" stroke-dasharray="5,3"/>
<line x1="15" y1="90" x2="275" y2="90" stroke="#30363d" stroke-width="1"/>
<text x="17" y="95" fill="#8b949e" font-size="9">0</text>
<text x="255" y="95" fill="#8b949e" font-size="9">5k</text>
<text x="17" y="17" fill="#58a6ff" font-size="9">α=0.200 (init)</text>
<text x="200" y="76" fill="#3fb950" font-size="9">α=0.050 (true)</text>
</svg>
</div>
<div class="result-plot" style="margin-bottom:12px;">
<div class="plot-title">EKI convergence — 15 iterations</div>
<svg viewBox="0 0 280 80" style="width:100%;height:80px;">
<path d="M15,15 C40,16 65,22 100,42 C130,55 170,62 220,64 C250,65 270,65 275,65" stroke="#ffa657" stroke-width="2" fill="none" marker-mid="url(#dot)"/>
<line x1="15" y1="65" x2="275" y2="65" stroke="#3fb950" stroke-width="1.5" stroke-dasharray="5,3"/>
<line x1="15" y1="72" x2="275" y2="72" stroke="#30363d" stroke-width="1"/>
<text x="17" y="70" fill="#8b949e" font-size="9">0</text>
<text x="262" y="70" fill="#8b949e" font-size="9">15</text>
<text x="195" y="60" fill="#3fb950" font-size="9">α_true=0.05</text>
</svg>
</div>
<div class="metric-row">
<div class="metric"><div class="val" style="color:var(--accent2);font-size:20px;">0.0502</div><div class="lbl">Gradient recovered α</div></div>
<div class="metric"><div class="val" style="color:var(--accent5);font-size:20px;">0.0497</div><div class="lbl">EKI recovered α</div></div>
<div class="metric"><div class="val" style="color:var(--muted);font-size:20px;">0.05</div><div class="lbl">True α</div></div>
<div class="metric"><div class="val" style="color:var(--accent);font-size:20px;">50</div><div class="lbl">Observations</div></div>
</div>
</div>
</div>
</div>
</div>
<!-- ========== S8: LESSON 08 UQ ========== -->
<div class="slide" id="s8">
<div class="slide-header"><span class="slide-tag t2">Lesson 08</span><div class="divider-line"></div><span class="slide-num">8 / 20</span></div>
<h2>Uncertainty Quantification — Three Methods</h2>
<div class="subtitle" style="margin-bottom:12px;">MC Dropout · Deep Ensemble · Conformal Prediction</div>
<div class="body">
<div class="three-col">
<div>
<div class="tier-block t1" style="margin-bottom:10px;">
<h4>MC Dropout</h4>
<pre style="font-size:11px;"><span class="kw">from</span> pinneaple_uq <span class="kw">import</span> (
<span class="cl">MCDropoutWrapper</span>,
<span class="cl">MCDropoutConfig</span>
)
cfg = <span class="cl">MCDropoutConfig</span>(
n_samples=<span class="nu">100</span>,
dropout_p=<span class="nu">0.05</span>,
)
wrap = <span class="cl">MCDropoutWrapper</span>(net, cfg)
uq = wrap.<span class="fn">predict_with_uncertainty</span>(x)
<span class="cm"># uq.mean, uq.std</span></pre>
</div>
<div class="card">
<h4>Properties</h4>
<ul class="bullets" style="font-size:12px;">
<li>Cheapest — modify existing net</li>
<li>No retraining needed</li>
<li>Coverage often underestimates</li>
</ul>
</div>
</div>
<div>
<div class="tier-block t2" style="margin-bottom:10px;">
<h4>Deep Ensemble</h4>
<pre style="font-size:11px;"><span class="kw">from</span> pinneaple_uq <span class="kw">import</span> (
<span class="cl">EnsembleUQ</span>,
<span class="cl">EnsembleConfig</span>
)
members = [<span class="fn">train_pinn</span>(seed=i)
<span class="kw">for</span> i <span class="kw">in</span> range(<span class="nu">5</span>)]
ens = <span class="cl">EnsembleUQ</span>(
models=members,
config=<span class="cl">EnsembleConfig</span>(n_members=<span class="nu">5</span>)
)
uq = ens.<span class="fn">predict_with_uncertainty</span>(x)</pre>
</div>
<div class="card">
<h4>Properties</h4>
<ul class="bullets" style="font-size:12px;">
<li>Most reliable uncertainty</li>
<li>Cost: 5× training</li>
<li>Diverse by different seeds</li>
</ul>
</div>
</div>
<div>
<div class="tier-block t3" style="margin-bottom:10px;">
<h4>Conformal Prediction</h4>
<pre style="font-size:11px;"><span class="kw">from</span> pinneaple_uq <span class="kw">import</span> <span class="cl">ConformalPredictor</span>
cp = <span class="cl">ConformalPredictor</span>(
model=net,
alpha=<span class="nu">0.10</span> <span class="cm"># 90% coverage</span>
)
cp.<span class="fn">calibrate</span>(x_cal, y_cal)
y_hat, lo, hi = cp.<span class="fn">predict</span>(x_test)
cov = cp.<span class="fn">coverage</span>(x_test, y_test)
<span class="cm"># Guaranteed: cov ≥ 1−α = 90%</span></pre>
</div>
<div class="card">
<h4>Properties</h4>
<ul class="bullets" style="font-size:12px;">
<li>Formal coverage guarantee</li>
<li>Post-hoc — no arch change</li>
<li>Requires calibration set</li>
</ul>
</div>
</div>
</div>
<div class="card">
<h4>Coverage results on heat equation (target marked in parentheses)</h4>
<table>
<tr><th>Method</th><th>Target</th><th>Empirical Coverage</th><th>Cost</th><th>Verdict</th></tr>
<tr><td>MC Dropout ±2σ</td><td>≈95%</td><td class="pp">88%</td><td class="better">1× (cheapest)</td><td class="xx">Slight undercover</td></tr>
<tr><td>Deep Ensemble ±2σ</td><td>≈95%</td><td class="better">94.2%</td><td class="worse">5× (expensive)</td><td class="better">Best reliability</td></tr>
<tr><td>Conformal (α=0.1)</td><td>≥90%</td><td class="better">91.5%</td><td class="better">1× (post-hoc)</td><td class="better">Guaranteed ✓</td></tr>
</table>
</div>
</div>
</div>
<!-- ========== S9: LESSON 09 TIME MARCHING ========== -->
<div class="slide" id="s9">
<div class="slide-header"><span class="slide-tag t2">Lesson 09</span><div class="divider-line"></div><span class="slide-num">9 / 20</span></div>
<h2>Time-Marching — Van der Pol Stiff Oscillator</h2>
<div class="subtitle" style="margin-bottom:12px;">x″ − μ(1−x²)x′ + x = 0 · μ=3 (stiff) · T=15 · Standard PINN fails</div>
<div class="body">
<div class="two-col">
<div>
<div class="hcard" style="margin-bottom:12px;">
<h4>TimeMarchingTrainer</h4>
<pre><span class="kw">from</span> pinneaple_train <span class="kw">import</span> <span class="cl">TimeMarchingTrainer</span>
<span class="cm"># Divide [0, 15] into 15 windows of 1 time unit</span>
<span class="cm"># Each window: fresh network + IC from previous end</span>
trainer = <span class="cl">TimeMarchingTrainer</span>(
model_factory = make_net, <span class="cm"># fresh per window</span>
loss_factory = make_loss, <span class="cm"># factory(ic, t0, t1)</span>
t_start = <span class="nu">0.0</span>,
t_end = <span class="nu">15.0</span>,
n_windows = <span class="nu">15</span>, <span class="cm"># 1s each</span>
epochs_per_window = <span class="nu">3_000</span>,
device = <span class="st">"cuda"</span>,
ic_state = torch.tensor([<span class="nu">2.0</span>, <span class="nu">0.0</span>]),
)
t_all, x_all = trainer.<span class="fn">integrate</span>()
<span class="cm"># Returns full-domain prediction by chaining windows</span></pre>
</div>
<div class="card">
<h4>Why standard PINN fails for μ=3</h4>
<ul class="bullets" style="font-size:12px;">
<li>Stiff ODE: fast and slow time scales coexist</li>
<li>PDE loss landscape very ill-conditioned over [0,15]</li>
<li>Gradient vanishes at early times when training whole domain</li>
<li>Time-marching: each 1s window is well-conditioned</li>
<li>CausalPINNTrainer is an alternative (causal weighting)</li>
</ul>
</div>
</div>
<div>
<div class="result-plot" style="margin-bottom:12px;">
<div class="plot-title">Van der Pol μ=3 — Standard PINN vs Time-Marching</div>
<svg viewBox="0 0 280 130" style="width:100%;height:140px;">
<!-- scipy reference — limit cycle -->
<path d="M15,65 C25,35 32,80 42,60 C50,40 58,85 68,60 C75,38 83,83 93,60 C100,40 108,83 118,60 C126,40 134,83 144,60 C152,40 160,83 170,60 C178,40 186,83 196,60 C204,40 212,83 222,60 C230,40 238,83 248,60 C256,40 264,75 270,60" stroke="#e6edf3" stroke-width="2" fill="none"/>
<!-- Standard PINN — diverges after t~5 -->
<path d="M15,65 C25,36 32,78 42,62 C50,42 58,82 68,62 C75,50 83,75 93,65 C100,60 108,70 118,68 C126,66 134,67 144,70 C152,72 160,75 170,80 C178,85 186,90 196,98 C204,105 212,110 240,118 C258,122 270,122 270,122" stroke="#d62728" stroke-width="1.5" fill="none" stroke-dasharray="5,3"/>
<!-- Time-marching — follows reference -->
<path d="M15,65 C25,36 32,79 42,61 C50,41 58,84 68,61 C75,39 83,83 93,60 C100,40 108,83 118,60 C126,40 134,83 144,60 C152,40 160,83 170,60 C178,40 186,83 196,60 C204,40 212,83 222,60 C230,40 238,83 249,60 C257,40 264,75 270,60" stroke="#3fb950" stroke-width="1.5" fill="none" stroke-dasharray="7,3"/>
<line x1="15" y1="125" x2="270" y2="125" stroke="#30363d" stroke-width="1"/>
<text x="16" y="130" fill="#8b949e" font-size="8">t=0</text>
<text x="252" y="130" fill="#8b949e" font-size="8">t=15</text>
<!-- Legend -->
<text x="16" y="12" fill="#e6edf3" font-size="8">scipy ref</text>
<text x="70" y="12" fill="#d62728" font-size="8">Standard PINN (fails)</text>
<text x="185" y="12" fill="#3fb950" font-size="8">TimeMarch</text>
</svg>
</div>
<div class="metric-row">
<div class="metric"><div class="val" style="color:var(--accent3);font-size:18px;">4.1e-1</div><div class="lbl">Standard PINN L2</div></div>
<div class="metric"><div class="val" style="color:var(--accent2);font-size:18px;">8.3e-3</div><div class="lbl">TimeMarch L2</div></div>
<div class="metric"><div class="val" style="color:var(--accent);font-size:18px;">15</div><div class="lbl">Time windows</div></div>
<div class="metric"><div class="val" style="color:var(--accent5);font-size:18px;">50×</div><div class="lbl">Error reduction</div></div>
</div>
</div>
</div>
</div>
</div>
<!-- ========== S10: LESSON 10 VALIDATION ========== -->
<div class="slide" id="s10">
<div class="slide-header"><span class="slide-tag t3">Lesson 10</span><div class="divider-line"></div><span class="slide-num">10 / 20</span></div>
<h2>Physics Validation — Never Skip This</h2>
<div class="subtitle" style="margin-bottom:12px;">Laplace ∇²u=0 · Well-trained vs under-trained · PhysicsValidator catches the bad one</div>
<div class="body">
<div class="two-col">
<div>
<div class="hcard" style="margin-bottom:12px;">
<h4>PhysicsValidator — fluent API</h4>
<pre><span class="kw">from</span> pinneaple_validate <span class="kw">import</span> (
<span class="cl">PhysicsValidator</span>, compare_to_analytical
)
validator = (
<span class="cl">PhysicsValidator</span>(
model=net,
coord_names=[<span class="st">"x"</span>, <span class="st">"y"</span>],
domain_bounds={<span class="st">"x"</span>:(<span class="nu">0</span>,<span class="nu">1</span>), <span class="st">"y"</span>:(<span class="nu">0</span>,<span class="nu">1</span>)},
)
.<span class="fn">add_boundary_check</span>(kind=<span class="st">"dirichlet"</span>, value=<span class="nu">0</span>,
boundary=<span class="st">"x=0"</span>, x_bc=bnd_pts)
.<span class="fn">add_boundary_check</span>(kind=<span class="st">"dirichlet"</span>, value_fn=g,
boundary=<span class="st">"x=1"</span>, x_bc=bnd_right)
)
report = validator.<span class="fn">validate</span>()
<span class="cm"># Three key metrics</span>
m = <span class="fn">compare_to_analytical</span>(
model=net, analytical_fn=exact,
coord_names=[<span class="st">"x"</span>,<span class="st">"y"</span>],
domain_bounds={...}, n_points=<span class="nu">20_000</span>,
)
<span class="cm"># m["rel_l2"], m["rmse"], m["max_error"]</span></pre>
</div>
<div class="card">
<h4>Why validate separately from training loss?</h4>
<ul class="bullets" style="font-size:12px;">
<li>Training loss = value at training points only</li>
<li>Validation uses 10–20× more held-out points</li>
<li>max_error reveals local failures (corners, BCs) that L2 hides</li>
<li>BC residual check: model may satisfy PDE but violate BC</li>
<li>Always run before claiming model is ready</li>
</ul>
</div>
</div>
<div>
<div class="card" style="margin-bottom:12px;">
<h4>Validation report — Well-trained vs Under-trained (Laplace)</h4>
<table>
<tr><th>Check</th><th>Well-trained</th><th>Under-trained</th></tr>
<tr><td>Relative L2</td><td class="better">4.2e-4</td><td class="worse">6.1e-2</td></tr>
<tr><td>RMSE</td><td class="better">3.1e-4</td><td class="worse">4.8e-2</td></tr>
<tr><td>Max error</td><td class="better">2.8e-3</td><td class="worse">2.4e-1</td></tr>
<tr><td>BC x=0 residual</td><td class="better">PASS 8.1e-5</td><td class="worse">FAIL 3.2e-2</td></tr>
<tr><td>BC x=1 residual</td><td class="better">PASS 1.2e-4</td><td class="worse">FAIL 4.7e-2</td></tr>
</table>
</div>
<div class="result-plot">
<div class="plot-title">|Error| field — left: well-trained · right: under-trained</div>
<svg viewBox="0 0 280 100" style="width:100%;height:100px;">
<defs>
<radialGradient id="gg" cx="50%" cy="50%" r="70%">
<stop offset="0%" stop-color="#3fb950" stop-opacity=".3"/>
<stop offset="100%" stop-color="#3fb950" stop-opacity=".0"/>
</radialGradient>
<radialGradient id="bb" cx="50%" cy="50%" r="70%">
<stop offset="0%" stop-color="#f78166" stop-opacity=".7"/>
<stop offset="100%" stop-color="#ffa657" stop-opacity=".2"/>
</radialGradient>
</defs>
<rect x="10" y="5" width="120" height="90" fill="url(#gg)" stroke="#3fb950" stroke-width="1" rx="4"/>
<rect x="150" y="5" width="120" height="90" fill="url(#bb)" stroke="#f78166" stroke-width="1" rx="4"/>
<text x="40" y="55" fill="#3fb950" font-size="14" font-weight="bold">PASS</text>
<text x="170" y="55" fill="#f78166" font-size="14" font-weight="bold">FAIL</text>
<text x="12" y="97" fill="#3fb950" font-size="8">L2=4.2e-4</text>
<text x="152" y="97" fill="#f78166" font-size="8">L2=6.1e-2</text>
</svg>
</div>
</div>
</div>
</div>
</div>
<!-- ========== S11: LESSON 11 OPERATORS ========== -->
<div class="slide" id="s11">
<div class="slide-header"><span class="slide-tag t3">Lesson 11</span><div class="divider-line"></div><span class="slide-num">11 / 20</span></div>
<h2>Operator Learning — Parametric Heat Family</h2>
<div class="subtitle" style="margin-bottom:12px;">Train on α ∈ {0.01…0.5} · Query any α instantly · No re-training</div>
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<h4>Parametric network: input (x, t, α)</h4>
<pre><span class="kw">from</span> pinneaple_models <span class="kw">import</span> <span class="cl">FourierFeatureEmbedding</span>
<span class="kw">class</span> <span class="cl">ParametricHeatNet</span>(nn.Module):
<span class="kw">def</span> <span class="fn">__init__</span>(self):
<span class="cm"># Spatial/temporal features: Fourier embed of (x,t)</span>
self.embed_xt = <span class="cl">FourierFeatureEmbedding</span>(
in_dim=<span class="nu">2</span>, n_fourier=<span class="nu">32</span>, sigma=<span class="nu">3.0</span>
)
<span class="cm"># Parameter encoder: α → 32-dim latent</span>
self.param_enc = nn.Sequential(
nn.Linear(<span class="nu">1</span>, <span class="nu">32</span>), nn.Tanh(),
nn.Linear(<span class="nu">32</span>, <span class="nu">32</span>), nn.Tanh(),
)
self.trunk = nn.Sequential(
nn.Linear(<span class="nu">64+32</span>, <span class="nu">128</span>), nn.Tanh(),
nn.Linear(<span class="nu">128</span>, <span class="nu">128</span>), nn.Tanh(),
nn.Linear(<span class="nu">128</span>, <span class="nu">1</span>),
)
<span class="kw">def</span> <span class="fn">forward</span>(self, x, t, alpha):
ft = self.embed_xt(cat([x, t], dim=-<span class="nu">1</span>))
fa = self.param_enc(alpha)
<span class="kw">return</span> self.trunk(cat([ft, fa], dim=-<span class="nu">1</span>))</pre>
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<h4>Key insight: PINN = one problem · Operator = many</h4>
<ul class="bullets" style="font-size:12px;">
<li>PINN: fixed α → needs re-training for new α (minutes/hours)</li>
<li>Operator: all α → forward pass for any α (milliseconds)</li>
<li>The parameter encoder learns a smooth α ↦ solution map</li>
<li>Interpolation between training α values works well</li>
<li>For larger families: use FNO (templates/19)</li>
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<div class="result-plot" style="margin-bottom:12px;">
<div class="plot-title">L2 error vs α — training (●) and test (✕)</div>
<svg viewBox="0 0 280 130" style="width:100%;height:130px;">
<line x1="30" y1="100" x2="270" y2="100" stroke="#30363d" stroke-width="1"/>
<line x1="30" y1="10" x2="30" y2="100" stroke="#30363d" stroke-width="1"/>
<!-- Training alphas: 0.01 0.02 0.05 0.1 0.2 0.5 mapped to x: 30 70 110 160 200 250 -->
<circle cx="45" cy="88" r="4" fill="#3fb950"/>
<circle cx="80" cy="85" r="4" fill="#3fb950"/>
<circle cx="118" cy="83" r="4" fill="#3fb950"/>
<circle cx="155" cy="85" r="4" fill="#3fb950"/>
<circle cx="195" cy="82" r="4" fill="#3fb950"/>
<circle cx="248" cy="88" r="4" fill="#3fb950"/>
<!-- Test alphas: 0.03 0.07 0.15 0.35 at 60 95 175 225 -->
<text x="58" y="78" fill="#ffa657" font-size="12">✕</text>
<text x="92" y="78" fill="#ffa657" font-size="12">✕</text>
<text x="170" y="78" fill="#ffa657" font-size="12">✕</text>
<text x="220" y="78" fill="#ffa657" font-size="12">✕</text>
<!-- Trend line -->
<path d="M30,85 Q140,82 270,87" stroke="#8b949e" stroke-width="1" fill="none" stroke-dasharray="3,2"/>
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<text x="32" y="112" fill="#8b949e" font-size="8">0.01</text>
<text x="110" y="112" fill="#8b949e" font-size="8">0.1</text>
<text x="245" y="112" fill="#8b949e" font-size="8">0.5</text>
<text x="12" y="90" fill="#8b949e" font-size="8">L2</text>
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<circle cx="35" cy="12" r="3" fill="#3fb950"/>
<text x="42" y="16" fill="#3fb950" font-size="8">Train α</text>
<text x="90" y="16" fill="#ffa657" font-size="8">✕ Test α</text>
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<h4>Results — train once, query any α</h4>
<table style="font-size:12px;">
<tr><th>α</th><th>Type</th><th>L2 error</th><th>Time</th></tr>
<tr><td>0.01</td><td class="better">Train</td><td class="better">4.1e-3</td><td>2 ms</td></tr>