ML for Science: Quick recap

01 Reverse-mode autodiff

The derivative of $F$ at $x$ is a linear map $dx \mapsto J\,dx$ . The JVP pushes a direction forward ( $v \mapsto Jv$ ); the VJP pulls a cotangent back ( $\bar y \mapsto J^\top \bar y$ ).
The VJP is defined by $\langle \bar y, J v\rangle = \langle \bar x, v\rangle$ for all $v$ . To derive one: write $dy$ , then move everything in $\langle \bar y, dy\rangle$ away from $dx$ .
Key VJPs: $C = AB$ gives $\bar A = \bar C B^\top$ , $\bar B = A^\top \bar C$ ; $X = A^{-1}B$ gives $\bar B = A^{-\top}\bar X$ , $\bar A = -\bar B X^\top$ ; $y = \log\lvert\det A\rvert$ gives $\bar A = \bar y\,A^{-\top}$ .
Reverse mode visits the tape in reverse order and adds the cotangents of a value that is used more than once (fan-out). It is back substitution on a triangular system.
A full $m \times n$ Jacobian needs $n$ forward passes or $m$ reverse passes, each a small constant times the cost of $F$ . For a scalar loss, one reverse pass gives the whole gradient.
Reverse mode pays in memory: the tape keeps every intermediate value a VJP needs.
Check gradients in float64 with central differences in a random direction, with $h \approx \varepsilon^{1/3} \approx 10^{-5}$ ; a bug shows as an error floor that does not move with $h$ .

02 Time stepping and stiffness

A one-step method has order $p$ if its local truncation error is $O(h^p)$ ; consistent + stable gives a global error $O(h^p)$ . Measure $p$ as the slope of $\log e$ against $\log h$ .
On the test equation $y' = \lambda y$ a method multiplies $y$ by $R(z)$ per step, $z = h\lambda$ . It is stable at $z$ if $\lvert R(z)\rvert \le 1$ .
$R(z) = 1 + z$ (forward Euler), $1/(1 - z)$ (backward Euler), $(1 + z/2)/(1 - z/2)$ (trapezoid / Crank–Nicolson). Forward Euler on the real axis: $-2 \le h\lambda \le 0$ .
Every explicit method has a bounded stability region. A-stable: the whole left half-plane is stable. L-stable: also $R(-\infty) = 0$ (backward Euler yes, Crank–Nicolson no).
For a system, every $h\lambda_i$ (eigenvalues of the Jacobian) must lie in the stability region. For explicit diffusion this gives $\Delta t \le \Delta x^2/(2\nu)$ in 1-D.
CFL condition: the numerical domain of dependence must contain the exact one; for upwind advection $\mathrm{Co} = c\Delta t/\Delta x \le 1$ . It is necessary, not sufficient.
A problem is stiff when stability, not accuracy, sets the step. Then use implicit, IMEX or exponential methods.

03 Adjoints and neural ODEs

The adjoint ODE is $\lambda' = -J_u^\top\lambda$ with the final condition $\lambda(T) = \nabla\ell(u(T))$ , solved backward in time; then $dL/du_0 = \lambda(0)$ and $dL/dp = \int_0^T J_p^\top\lambda\,dt$ .
The adjoint method costs about one extra (backward) solve for any number of parameters; finite differences cost $2n_p$ solves and forward sensitivities ( $S' = J_uS + J_p$ ) about $n_p$ .
A residual network $x_{k+1} = x_k + h f(x_k, \theta)$ is forward Euler for the neural ODE $u' = f_\theta(u)$ ; the adjoint method is backpropagation in time.
The discrete adjoint (backprop through the solver steps, discretize-then-optimize) is the exact gradient of the computed loss; the continuous adjoint (optimize-then-discretize) differs from it by $O(h^q)$ .
Backsolve re-creates $u$ backward with $O(1)$ memory, but every mode that decays forward grows backward, so it fails for damped or stiff problems; the adjoint equation itself is stable.
Uniform checkpointing stores $C$ states and recomputes the rest; memory $\approx C + N/C$ is smallest at $C = \sqrt N$ , for less than one extra forward pass, and the gradient is unchanged.
ODE trajectories cannot cross, so a 1D neural ODE can only learn increasing maps; augmented neural ODEs add dimensions.

04 Inverse problems and seismic imaging

An inverse problem recovers a model $m$ from data $d = F(m) + e$ . It is ill-posed when a solution fails to exist, to be unique, or to depend stably on the data (Hadamard); most inverse problems in science fail stability.
For linear $G = \sum_i \sigma_i u_iv_i^\top$ , the naive solution $\sum_i (u_i^\top d/\sigma_i)\,v_i$ divides the noise by $\sigma_i$ , so small singular values make it useless; the discrete Picard condition says $|u_i^\top d|$ must decay faster than $\sigma_i$ .
Tikhonov regularization $\min\|Gm - d\|^2 + \alpha^2\|m\|^2$ multiplies each component by the filter factor $f_i = \sigma_i^2/(\sigma_i^2 + \alpha^2)$ , which is $\tfrac12$ at $\sigma_i = \alpha$ ; truncated SVD uses $f_i \in \{0, 1\}$ .
Choose $\alpha$ with the discrepancy principle ( $\|Gm_\alpha - d\| \approx \sqrt{n_d}\,\sigma$ ), the corner of the L-curve, or GCV.
Tikhonov is the MAP estimate for noise $\mathcal N(0, \sigma^2 I)$ and prior $\mathcal N(0, (\sigma^2/\alpha^2) I)$ ; the posterior covariance is $\sigma^2(G^\top G + \alpha^2 I)^{-1}$ , and with general Gaussians the cost is 3D-Var.
Gauss–Newton solves $(J_F^\top J_F)\delta = -J_F^\top r$ at each step; in earthquake location the station geometry sets the conditioning of $J_F^\top J_F$ and the shape of the uncertainty ellipse.
The adjoint-state method gives $\partial J/\partial m_k = \lambda^\top(\partial A/\partial m_k)u$ with $A^\top\lambda = -\nabla_u j$ : one forward and one adjoint solve per source, for any number of parameters. For FWI, $\nabla_m J = \int_0^T\lambda\,u_{tt}\,dt$ .
FWI cycle-skips when the predicted arrivals are wrong by more than about half a period; inverting low frequencies first widens the basin of the correct model.

05 Chaos, DMD and SINDy

A fixed point is classified by the eigenvalues of the Jacobian there: real parts decide growth or decay, imaginary parts decide rotation. Lorenz-63 has three fixed points, and all are unstable.
The Lyapunov exponents are $\lambda_i = \lim_{t\to\infty}\frac1t\ln s_i(M(t))$ for the tangent propagator $M$ . A bounded system with $\lambda_1 > 0$ is chaotic; for Lorenz-63, $\lambda = (0.906, 0, -14.572)$ .
The sum of the Lyapunov exponents is the time average of $\operatorname{tr}J$ (Liouville's formula), and a flow that does not stop at a fixed point has one zero exponent.
An error reaches the size $\Delta$ after $T_p = \lambda_1^{-1}\ln(\Delta/\delta_0)$ : a 10 times smaller starting error adds only $\ln(10)/\lambda_1$ of forecast time.
The Koopman operator $(\mathcal Kg)(u) = g(F(u))$ is linear on observables even when $F$ is not; a Koopman eigenfunction evolves as $\varphi(u_k) = \mu^k\varphi(u_0)$ . Chaotic systems have no exact finite Koopman-invariant subspace.
Exact DMD fits $A = X'X^+$ through a rank- $r$ SVD without forming $A$ ; its modes are eigenvectors of $A$ , and each eigenvalue $\mu$ gives the growth rate $\ln|\mu|/\Delta t$ and the angular frequency $\arg\mu/\Delta t$ .
SINDy writes $\dot X \approx \Theta(X)\Xi$ with a sparse $\Xi$ and finds it by sequentially thresholded least squares; the threshold must lie between the largest spurious and the smallest true coefficient.

06 Data assimilation

Data assimilation is Bayes' rule: the analysis (posterior) is proportional to forecast (prior) × observation likelihood. For one variable, $x^a = x^f + K(y - x^f)$ with $K = P^f/(P^f + R)$ and $1/P^a = 1/P^f + 1/R$ .
The Kalman update follows from Gaussian conditioning: $K = P^fH^\top(HP^fH^\top + R)^{-1}$ , $x^a = x^f + K(y - Hx^f)$ , $P^a = (I - KH)P^f$ . $K$ and $P^a$ do not depend on the value of $y$ .
An unobserved variable changes only through its forecast covariance with the observed quantities.
The Kalman filter alternates the forecast step $x^f_k = Mx^a_{k-1}$ , $P^f_k = MP^a_{k-1}M^\top + Q$ and the analysis step; it is exact for linear-Gaussian models, but it is too costly for $n \sim 10^8$ .
The stochastic EnKF uses the sample covariance of $N$ members and perturbed observations (without them the analysis spread misses $KRK^\top$ ); its increments lie in at most $N - 1$ directions.
Localisation ( $\rho \circ P^f$ ) removes spurious long-range correlations and raises the rank; inflation ( $\lambda^2P^f$ ) fixes underdispersion but keeps the span and the correlations.
3D-Var and 4D-Var compute the MAP estimate; for linear models the 3D-Var minimum is the Kalman mean and the Hessian is $(P^a)^{-1}$ .
The 4D-Var gradient is $B^{-1}(x_0 - x^b) + \lambda_0$ from one backward sweep $\lambda_j = M_j^\top\lambda_{j+1} + f_j$ : one forward and one adjoint run, whatever the size of the state.

07 Gaussian processes and UQ

A Gaussian process is a distribution over functions: any finite set of values is jointly Gaussian with covariance $k(x_i, x_j)$ . The length scale $\ell$ sets how fast the function varies; $\sigma_f^2$ sets its size.
GP posterior: $f_* \mid y \sim \mathcal N(K_*^\top K^{-1}y,\; K_{**} - K_*^\top K^{-1}K_*)$ with $K = K_f + \sigma_n^2 I$ . It is Gaussian conditioning, and its covariance does not depend on $y$ .
A new measurement $y_*$ has variance $\operatorname{Var}[f_*] + \sigma_n^2$ ; intervals without $\sigma_n^2$ are too narrow.
Compute with one Cholesky factor $K = LL^\top$ : $\alpha = L^\top\backslash(L\backslash y)$ , $\Sigma_* = K_{**} - V^\top V$ with $V = L\backslash K_*$ , and $\log\det K = 2\sum_i\log L_{ii}$ . Never form $K^{-1}$ or $\det K$ .
Log marginal likelihood $= -\tfrac12 y^\top K^{-1}y - \tfrac12\log\det K - \tfrac n2\log 2\pi$ (data fit, complexity, constant). Its gradient is $\tfrac12\operatorname{tr}((\alpha\alpha^\top - K^{-1})\,\partial K/\partial\theta_j)$ ; it can have several local maxima.
A scoring rule is proper when the honest forecast minimises the expected score. The CRPS $= \mathbb E|X - y| - \tfrac12\mathbb E|X - X'|$ is strictly proper, has the units of $y$ , and equals the absolute error for a point forecast.
Calibration: PIT values $F(y)$ are uniform; a U shape means too narrow, a hump means too wide. Coverage checks several levels. The aim is maximum sharpness subject to calibration.

08 Spectral methods and FNO

On a periodic grid, $d/dx$ is multiplication by $ik$ in Fourier space; the spectral derivative is $\operatorname{ifft}(ik\,\operatorname{fft}(u))$ , with the Nyquist mode set to zero for odd derivatives.
Spectral accuracy: for analytic periodic functions the error falls like $e^{-an/2}$ , faster than any power of $n$ ; with finitely many derivatives only like a power of $1/n$ .
Aliasing: on $n$ points, modes $k$ and $k + qn$ have the same samples, so $\hat u_k/n = \sum_q c_{k+qn}$ .
Orszag's 2/3 rule: for a quadratic term, keep $\lvert k\rvert \le K$ with $3K < n$ ; then the pseudo-spectral product is exact on the kept modes.
Dealiased pseudo-spectral Burgers conserves energy for $\nu = 0$ and dissipates it for $\nu > 0$ , like the PDE; aliasing can create energy and blow up an under-resolved run.
Cole–Hopf, $u = -2\nu\phi_x/\phi$ , turns viscous Burgers into the heat equation and gives exact test solutions.
An FNO Fourier layer multiplies the lowest $k_{\max}$ modes by learned matrices $R_k$ ; its weights do not depend on $n$ .
The Fourier layer is exactly discretisation-invariant for band-limited input; pointwise activations alias, so the full FNO is only approximately invariant.

09 Spherical harmonics and SFNO

A latitude–longitude grid is not a periodic image: latitude is not periodic, and the rows near the poles hold many points on a small area.
The real spherical harmonics $Y_l^m$ ( $l \ge 0$ , $-l \le m \le l$ ) are orthonormal on the sphere; $Y_l^m$ has $|m|$ meridian nodal lines and $l - |m|$ latitude nodal lines.
Spherical harmonics are eigenfunctions of the Laplacian: $\nabla^2 Y_l^m = -l(l+1)Y_l^m$ on the unit sphere, $-l(l+1)/a^2$ on radius $a$ ; so derivatives, Poisson solves and diffusion are multiplications per degree.
The triangular truncation $T_N$ has $(N + 1)^2$ real coefficients and the same resolution everywhere on the sphere.
Compute $\bar P_l^m$ with the normalised three-term recursion, never with factorials.
A $K$ -node Gauss–Legendre rule is exact up to degree $2K - 1$ ; with $N + 1$ Gauss latitudes and $2N + 1$ longitudes, the SHT (FFT in longitude, then Legendre quadrature) is exact for $T_N$ fields.
A rotation maps the degree- $l$ coefficients by an orthogonal matrix $D^l(R)$ ; so the power spectrum $E_l$ is rotation-invariant, and a filter $c_l$ that depends only on $l$ commutes with rotations.
The SFNO layer is the FNO layer with the SHT: a learned channel matrix $R_l$ per degree, independent of $m$ , plus a pointwise skip and activation.

10 Equivariant networks

A map is equivariant if $f(\rho_{\text{in}}(g)x) = \rho_{\text{out}}(g)f(x)$ for all $g$ ; energy is E(3)-invariant and forces are E(3)-equivariant, because the gradient of an invariant function is equivariant.
Translations are handled by using only relative positions $x_j - x_i$ ; the real work is the group $O(3)$ = rotations plus the inversion.
The irreps of $SO(3)$ are the degrees $l = 0, 1, 2, \dots$ with dimension $2l + 1$ ; $O(3)$ adds a parity: 0e scalar, 0o pseudoscalar, 1o vector, 1e pseudovector, 2e symmetric traceless matrix.
The real Wigner-D matrix is defined by $Y_l(R\hat x) = D^l(R)Y_l(\hat x)$ ; it is an orthogonal representation, and $D^1(R) = R$ in the $(x, y, z)$ order.
Schur's lemma: an equivariant linear map between different irreps is zero, and between equal (real $SO(3)$ ) irreps it is $\lambda I$ ; so equivariant linear layers only mix channels within each degree and parity.
Clebsch–Gordan: $D^{l_1} \otimes D^{l_2} = \bigoplus_{L=|l_1-l_2|}^{l_1+l_2} D^L$ , parities multiply, and two vectors give $1 \otimes 1 = 0 \oplus 1 \oplus 2$ (dot, cross, symmetric traceless).
Nonlinearities act on scalars; higher degrees are multiplied by invariant gates $\sigma(s)$ or by functions of their norm.
Test equivariance with random $Q \in O(3)$ of both determinants and random shifts; an exact model gives errors at rounding level, and the test must be able to fail.

11 PINNs and failure modes

PINN loss: $L = \frac1{N_r}\sum_i r_\theta(x_i, t_i)^2 + \lambda_{ic}\frac1{N_0}\sum_j(u_\theta(x_j, 0) - g(x_j))^2 + \lambda_{bc}L_{bc}$ , with the residual computed by automatic differentiation with respect to the inputs.
Second derivatives need nested reverse sweeps (create_graph=True) and a smooth activation; ReLU gives $u_{xx} = 0$ .
For advection, $\lVert e(t)\rVert \le \lVert e(0)\rVert + \int_0^t\lVert r(s)\rVert\,ds$ : a small residual guarantees a small error, so a failed PINN is a training failure, not a capacity failure.
At large $\beta$ the residual term grows like $\beta^2$ and the zero field has a far lower loss ( $\lambda_{ic}/2$ ) than the frozen initial condition ( $\beta^2/2$ ).
Spectral bias: gradient descent fits low frequencies first, so fast oscillations in $t$ are learned slowly.
Fixes: curriculum in $\beta$ , time marching over short windows, and causal weights $w_i = \exp(-\varepsilon\sum_{k<i}L_r(t_k))$ treated as constants.
For a clean forward problem a classical solver is far more accurate and cheaper; PINNs suit inverse problems with sparse data and a known PDE.
Data misfit plus residual penalty is weak-constraint 4D-Var with the network as the trajectory.

12 SDEs and Fokker–Planck

Brownian motion has independent increments $W_{t+h} - W_t \sim \mathcal N(0, h)$ . Its quadratic variation $\sum(\Delta W)^2$ tends to $T$ while its total variation is infinite; in short, $(dW)^2 = dt$ .
Euler–Maruyama: $X_{n+1} = X_n + a\,h + b\,\Delta W_n$ with the coefficients at the left point; this defines the Itô SDE $dX = a\,dt + b\,dW$ .
Itô's lemma: $f(X, t)$ is an Itô process with drift $f_t + a f_x + \tfrac12 b^2 f_{xx}$ and diffusion coefficient $b f_x$ . The term $\tfrac12 b^2f_{xx}$ is the Itô correction.
GBM $dX = \mu X\,dt + \sigma X\,dW$ has $X_t = x_0e^{(\mu - \sigma^2/2)t + \sigma W_t}$ : mean growth rate $\mu$ , median growth rate $\mu - \sigma^2/2$ .
The OU process $dX = -\theta X\,dt + \sigma\,dW$ is Gaussian with mean $x_0e^{-\theta t}$ and variance $\frac{\sigma^2}{2\theta}(1 - e^{-2\theta t})$ ; its stationary law is $\mathcal N(0, \sigma^2/2\theta)$ .
Fokker–Planck: $\partial_t p = -\partial_x(ap) + \tfrac12\partial_{xx}(b^2p) = -\partial_x J$ ; stationary densities have zero flux, and $dX = -U'\,dt + \sqrt2\,dW$ has the Gibbs density $p \propto e^{-U}$ .
Euler–Maruyama has strong order 1/2 (1 for additive noise) and weak order 1; Milstein adds $\tfrac12 b b_x((\Delta W)^2 - h)$ and has strong order 1.

13 Score matching and diffusion

The score $\nabla_x\log p$ points toward high density and does not depend on the normalising constant of $p$ .
Denoising score matching: $J_{\text{DSM}} = E\|s_\theta(x_0 + \sigma\varepsilon) + \varepsilon/\sigma\|^2 = J_{\text{ESM}} + C$ , because the marginal score is the posterior mean of the conditional score; the minimum $C > 0$ is not zero.
Tweedie's formula: $E[x_0 \mid x] = x + \sigma^2\nabla\log p_\sigma(x)$ ; learning to denoise and learning the score are the same task.
Noising SDE $dx = f\,dt + g\,dW$ : VE ( $f = 0$ , here $g^2 = 2t$ , noise level $\sigma = t$ ) and VP (an OU process); the marginals $p_t$ are the data convolved with a Gaussian.
Reverse-time SDE (Anderson 1982): $dx = [f - g^2\nabla\log p_t]\,dt + g\,d\bar W$ , run from $t = T$ to $0$ , has the same marginals $p_t$ .
Probability-flow ODE: $dx/dt = f - \tfrac12 g^2\nabla\log p_t$ has the same marginals; it is deterministic, invertible and gives exact likelihoods.
Both samplers follow from writing the Fokker–Planck equation as a continuity equation; any noise level $\tilde g$ between them also works.
In practice: Euler–Maruyama for the SDE, Heun (second order) for the ODE, a geometric grid in $\sigma$ , and a start from $\mathcal N(0, T^2I)$ .

14 Flow matching and OT

Flow matching learns the velocity $u_t$ of an ODE from noise ( $t = 0$ ) to data ( $t = 1$ ) by regression on the conditional target $x_1 - x_0$ along straight lines $x_t = (1 - t)x_0 + tx_1$ ; no ODE is solved during training.
Marginalization trick: $u_t(x) = \mathbb E[x_1 - x_0 \mid x_t = x]$ generates the path $p_t$ , and the CFM loss equals the FM loss plus a constant $V \ge 0$ (the conditional variance), so both have the same gradients.
With an independent coupling, lines cross and the learned paths bend; for noise $\mathcal N(0, 1)$ and data $\pm 2$ one Euler step sends every sample to $0$ .
Rectified flow (re-pairing $Z_0$ with the flow's own end point $Z_1$ ) does not increase any convex transport cost; for the squared cost the drop is $S + V$ , and reflow straightens the paths.
1D optimal transport is sorting; for Gaussians, $W_2^2 = \lVert m_1 - m_2\rVert^2 + \operatorname{tr}\big(S_1 + S_2 - 2(S_1^{1/2}S_2S_1^{1/2})^{1/2}\big)$ (Bures).
Entropic OT has the plan $P_{ij} = e^{(f_i + g_j - C_{ij})/\varepsilon}$ ; Sinkhorn alternates row and column scalings (log domain for small $\varepsilon$ ), and its cost is above the exact one by at most $\varepsilon\min(H(a), H(b))$ .
Pairs of an optimal plan for the squared cost never meet at the same time, so OT paths are straight and one Euler step is exact; minibatch OT moves the coupling part of the way toward this.

15 Flow matching on the sphere

The tangent plane of $S^2$ at $x$ is $\{v : x \cdot v = 0\}$ , with the projection $P_x = I - xx^\top$ ; the geodesic distance is $d(x, y) = \arccos(x \cdot y)$ , and geodesics are great circles at constant speed.
$\exp_x(v) = \cos\lVert v\rVert\,x + \sin\lVert v\rVert\,v/\lVert v\rVert$ , and $\log_x(y) = \omega\,u/\lVert u\rVert$ with $\omega = \arccos(x\cdot y)$ and $u = y - (x\cdot y)x$ ; they are inverse for $\lVert v\rVert < \pi$ .
$\log_x$ is undefined at the antipode $-x$ (the cut locus of $x$ ), and its direction jumps near it.
The geodesic conditional path $x_t = \exp_{x_0}(t\log_{x_0}x_1)$ is slerp, and its velocity is $\log_{x_t}(x_1)/(1 - t)$ , with constant speed $d(x_0, x_1)$ .
The Riemannian CFM loss regresses a tangent model $P_x\operatorname{net}_\theta$ on $\log_{x_t}(x_1)/(1 - t)$ ; it has the same gradient as Riemannian flow matching, needs no simulation on the sphere, and is the CFM of lesson 14 in $\R^d$ .
Plain Euler steps leave the sphere ( $\lVert x + hv\rVert^2 = 1 + h^2\lVert v\rVert^2$ ); exponential-map steps or projected steps (a retraction) keep samples on it.

16 ML weather and climate models

An ML weather model learns the map $x_t \to x_{t + 6\text{h}}$ from reanalysis data (ERA5, made with 4D-Var) and rolls it out to 10–15 days; families differ in how they move information on the sphere (mesh GNNs, 3D transformers, SFNO, hybrid physics–ML, diffusion ensembles).
Rollout fine-tuning trains on several autoregressive steps with the gradient through all of them, so the model learns to handle its own errors; it cannot remove chaotic error growth.
Forecast errors grow with lead time and level off: a single forecast near $\sqrt2\,\sigma_c$ , an ensemble mean near the climatological $\sigma_c$ .
Theorem 1: $\mathbb E[(f - o)^2 \mid \mathcal I] = (f - \mu)^2 + \operatorname{Var}(o \mid \mathcal I)$ , so MSE training produces the smooth conditional mean, and a sharp realistic forecast pays the double penalty (twice the floor).
On a latitude–longitude grid, weight each row by its area, $w_i \propto \sin\phi_i^u - \sin\phi_i^l \approx \cos\phi_i$ ; poleward of 60° are 33.6 % of the rows of the 0.25° grid but 13.4 % of the area.
The fair ensemble CRPS $\frac1M\sum|x_m - y| - \frac{1}{2M(M-1)}\sum_{m,n}|x_m - x_n|$ is unbiased for the CRPS of the member distribution; the standard $1/(2M^2)$ version punishes small ensembles.
Theorem 2: for a perfect $M$ -member ensemble, skill² $= \frac{M+1}{M}$ spread², so spread/skill $\approx \sqrt{M/(M+1)}$ (0.953 for $M = 10$ ).
A fair comparison fixes the test year (WB2: 2020), the ground truth, the grid and regridding, the time convention and the units, and tests differences with paired scores and a block bootstrap.