build based on d9fd208

dev/.documenter-siteinfo.json

@@ -1 +1 @@
-{"documenter":{"julia_version":"1.10.3","generation_timestamp":"2024-05-17T01:24:53","documenter_version":"1.4.1"}}
+{"documenter":{"julia_version":"1.10.3","generation_timestamp":"2024-05-24T01:24:48","documenter_version":"1.4.1"}}

dev/DeepBSDE/index.html

@@ -64,4 +64,4 @@
  trajectories_lower,
  maxiters_limits
 )
-</code></pre><p>Returns a <code>PIDESolution</code> object. </p><p><strong>Arguments:</strong></p><ul><li><code>maxiters</code>: The number of training epochs. Defaults to <code>300</code></li><li><code>trajectories</code>: The number of trajectories simulated for training. Defaults to <code>100</code></li></ul><p>To use <a href="https://diffeq.sciml.ai/stable/solvers/sde_solve/">SDE Algorithms</a> use <a href="../tutorials/deepbsde/#DeepBSDE"><code>DeepBSDE</code></a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/HighDimPDE.jl/blob/d9fd208c79bd49728e5eb10a93fa2d539acfcc74/src/DeepBSDE_Han.jl#L2">source</a></section></article><h2 id="The-general-idea"><a class="docs-heading-anchor" href="#The-general-idea">The general idea 💡</a><a id="The-general-idea-1"></a><a class="docs-heading-anchor-permalink" href="#The-general-idea" title="Permalink"></a></h2><p>The <code>DeepBSDE</code> algorithm is similar in essence to the <code>DeepSplitting</code> algorithm, with the difference that it uses two neural networks to approximate both the the solution and its gradient.</p><h2 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h2><ul><li>Han, J., Jentzen, A., E, W., Solving high-dimensional partial differential equations using deep learning. <a href="https://arxiv.org/abs/1707.02568">arXiv</a> (2018)</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../DeepSplitting/">« The <code>DeepSplitting</code> algorithm</a><a class="docs-footer-nextpage" href="../NNStopping/">The <code>NNStopping</code> algorithm »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Friday 17 May 2024 01:24">Friday 17 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+</code></pre><p>Returns a <code>PIDESolution</code> object. </p><p><strong>Arguments:</strong></p><ul><li><code>maxiters</code>: The number of training epochs. Defaults to <code>300</code></li><li><code>trajectories</code>: The number of trajectories simulated for training. Defaults to <code>100</code></li></ul><p>To use <a href="https://diffeq.sciml.ai/stable/solvers/sde_solve/">SDE Algorithms</a> use <a href="../tutorials/deepbsde/#DeepBSDE"><code>DeepBSDE</code></a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/HighDimPDE.jl/blob/d9fd208c79bd49728e5eb10a93fa2d539acfcc74/src/DeepBSDE_Han.jl#L2">source</a></section></article><h2 id="The-general-idea"><a class="docs-heading-anchor" href="#The-general-idea">The general idea 💡</a><a id="The-general-idea-1"></a><a class="docs-heading-anchor-permalink" href="#The-general-idea" title="Permalink"></a></h2><p>The <code>DeepBSDE</code> algorithm is similar in essence to the <code>DeepSplitting</code> algorithm, with the difference that it uses two neural networks to approximate both the the solution and its gradient.</p><h2 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h2><ul><li>Han, J., Jentzen, A., E, W., Solving high-dimensional partial differential equations using deep learning. <a href="https://arxiv.org/abs/1707.02568">arXiv</a> (2018)</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../DeepSplitting/">« The <code>DeepSplitting</code> algorithm</a><a class="docs-footer-nextpage" href="../NNStopping/">The <code>NNStopping</code> algorithm »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Friday 24 May 2024 01:24">Friday 24 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

dev/Feynman_Kac/index.html

@@ -7,4 +7,4 @@
 v(\tau, x) &= \int_{-\tau}^0 \mathbb{E} \left[ f(X^x_{s + \tau}, v(s + T, X^x_{s + \tau}))ds \right] + \mathbb{E} \left[ v(0, X^x_{\tau}) \right]\\
  &= - \int_{\tau}^0 \mathbb{E} \left[ f(X^x_{\tau - s}, v(T-s, X^x_{\tau - s}))ds \right] + \mathbb{E} \left[ v(0, X^x_{\tau}) \right]\\
  &= \int_{0}^\tau \mathbb{E} \left[ f(X^x_{\tau - s}, v(T-s, X^x_{\tau - s}))ds \right] + \mathbb{E} \left[ v(0, X^x_{\tau}) \right].
-\end{aligned}\]</p><p>This leads to the </p><div class="admonition is-info"><header class="admonition-header">Non-linear Feynman Kac for initial value problems</header><div class="admonition-body"><p>Consider the PDE</p><p class="math-container">\[\partial_t u(t,x) = \mu(t, x) \nabla_x u(t,x) + \frac{1}{2} \sigma^2(t, x) \Delta_x u(t,x) + f(x, u(t,x))\]</p><p>with initial conditions <span>$u(0, x) = g(x)$</span>, where <span>$u \colon \R^d \to \R$</span>. Then</p><p class="math-container">\[u(t, x) = \int_0^t \mathbb{E} \left[ f(X^x_{t - s}, u(T-s, X^x_{t - s}))ds \right] + \mathbb{E} \left[ u(0, X^x_t) \right] \tag{3}\]</p><p>with </p><p class="math-container">\[X_t^x = \int_0^t \mu(X_s^x)ds + \int_0^t\sigma(X_s^x)dB_s + x.\]</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../tutorials/nnparamkolmogorov/">« <code>NNParamKolmogorov</code></a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Friday 17 May 2024 01:24">Friday 17 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\end{aligned}\]</p><p>This leads to the </p><div class="admonition is-info"><header class="admonition-header">Non-linear Feynman Kac for initial value problems</header><div class="admonition-body"><p>Consider the PDE</p><p class="math-container">\[\partial_t u(t,x) = \mu(t, x) \nabla_x u(t,x) + \frac{1}{2} \sigma^2(t, x) \Delta_x u(t,x) + f(x, u(t,x))\]</p><p>with initial conditions <span>$u(0, x) = g(x)$</span>, where <span>$u \colon \R^d \to \R$</span>. Then</p><p class="math-container">\[u(t, x) = \int_0^t \mathbb{E} \left[ f(X^x_{t - s}, u(T-s, X^x_{t - s}))ds \right] + \mathbb{E} \left[ u(0, X^x_t) \right] \tag{3}\]</p><p>with </p><p class="math-container">\[X_t^x = \int_0^t \mu(X_s^x)ds + \int_0^t\sigma(X_s^x)dB_s + x.\]</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../tutorials/nnparamkolmogorov/">« <code>NNParamKolmogorov</code></a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Friday 24 May 2024 01:24">Friday 24 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

dev/MLP/index.html

@@ -16,4 +16,4 @@
 u_L &= \sum_{l=1}^{L-1} \frac{1}{M^{L-l}}\sum_{i=1}^{M^{L-l}} \frac{1}{K}\sum_{j=1}^{K} \bigg[ f(X^{x,(l, i)}_{t - s_{(l, i)}}, Z^{(l,j)}, u(T-s_{(l, i)}, X^{x,(l, i)}_{t - s_{(l, i)}}), u(T-s_{l,i}, Z^{(l,j)})) + \\
 &\qquad 
 \mathbf{1}_\N(l) f(X^{x,(l, i)}_{t - s_{(l, i)}}, u(T-s_{(l, i)}, X^{x,(l, i)}_{t - s_{(l, i)}}))\bigg] + \frac{1}{M^{L}}\sum_i^{M^{L}} u(0, X^{x,(l, i)}_t)\\
-\end{aligned}\]</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>In practice, if you have a non-local model, you need to provide the sampling method and the number <span>$K$</span> of MC integration through the keywords <code>mc_sample</code> and <code>K</code>. </p><ul><li><code>K</code> characterizes the number of samples for the Monte Carlo approximation of the last term.</li><li><code>mc_sample</code> characterizes the distribution of the <code>Z</code> variables</li></ul></div></div><h2 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h2><ul><li>Boussange, V., Becker, S., Jentzen, A., Kuckuck, B., Pellissier, L., Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions. <a href="https://arxiv.org/abs/2205.03672">arXiv</a> (2022)</li><li>Becker, S., Braunwarth, R., Hutzenthaler, M., Jentzen, A., von Wurstemberger, P., Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations. <a href="https://arxiv.org/abs/2005.10206">arXiv</a> (2020)</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../problems/">« Problems</a><a class="docs-footer-nextpage" href="../DeepSplitting/">The <code>DeepSplitting</code> algorithm »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Friday 17 May 2024 01:24">Friday 17 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\end{aligned}\]</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>In practice, if you have a non-local model, you need to provide the sampling method and the number <span>$K$</span> of MC integration through the keywords <code>mc_sample</code> and <code>K</code>. </p><ul><li><code>K</code> characterizes the number of samples for the Monte Carlo approximation of the last term.</li><li><code>mc_sample</code> characterizes the distribution of the <code>Z</code> variables</li></ul></div></div><h2 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h2><ul><li>Boussange, V., Becker, S., Jentzen, A., Kuckuck, B., Pellissier, L., Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions. <a href="https://arxiv.org/abs/2205.03672">arXiv</a> (2022)</li><li>Becker, S., Braunwarth, R., Hutzenthaler, M., Jentzen, A., von Wurstemberger, P., Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations. <a href="https://arxiv.org/abs/2005.10206">arXiv</a> (2020)</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../problems/">« Problems</a><a class="docs-footer-nextpage" href="../DeepSplitting/">The <code>DeepSplitting</code> algorithm »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Friday 24 May 2024 01:24">Friday 24 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>