https://www.chenk.top/en/tags/automatic-differentiation/Recent content in Automatic Differentiation on Chen Kai BlogHugoenWed, 01 May 2024 09:00:00 +0000https://www.chenk.top/en/pde-ml/01-physics-informed-neural-networks/Wed, 01 May 2024 09:00:00 +0000https://www.chenk.top/en/pde-ml/01-physics-informed-neural-networks/<blockquote> <p><strong>Series chapter 1 — about a 35-minute read.</strong> This is the foundation of the entire series. Neural operators, variational principles, score matching — every later chapter is, at heart, <em>the same idea</em>: how to encode physical or mathematical constraints directly into the neural network&rsquo;s optimization objective. Master PINNs, and the rest is just swapping one constraint for another.</p> </blockquote> <hr> <p><figure class="article-figure"> <img src="https://blog-pic-ck.oss-cn-beijing.aliyuncs.com/posts/en/pde-ml/01-Physics-Informed-Neural-Networks/illustration_1.png" alt="PDE and ML (1): Physics-Informed Neural Networks — Chapter overview" loading="lazy" decoding="async" class="content-image"> </figure> </p> <h2 id="prologue-a-metal-rod" class="heading-anchor">Prologue: a metal rod<a href="#prologue-a-metal-rod" class="heading-link" aria-label="Permalink to this section" title="Copy link to this section">#</a> </h2><p>Suppose you want the temperature distribution <span class="math-inline">$u(x,t)$</span> along a metal rod. Half a century of numerical analysis offers two standard answers:</p>