---
title: 'LaSDI-IT: Latent Dynamics for Interface Tracking'
url: https://www.emergentmind.com/topics/lasdi-it
type: topic
---

# LaSDI-IT: Latent Dynamics for Interface Tracking

LaSDI-IT, short for **Latent Space Dynamics Identification for Interface Tracking**, is a physics-informed, data-driven reduced-order modeling framework for systems with sharp, moving interfaces such as shocks, phase boundaries, fracture fronts, and void or pore boundaries. It extends the LaSDI and GPLaSDI lineage by modifying the autoencoder and training objective so that the reduced model does not only reconstruct the physical field, but also explicitly learns the location of the interface. In the reported shock-induced pore-collapse study, LaSDI-IT combines interface-aware encoding, linear latent dynamics, Gaussian-process interpolation, and greedy sampling to obtain relative prediction errors below \(9\%\) across the parameter space while recovering quantities of interest such as pore area and hot spot formation [2507.10647].

## 1. Motivation and problem setting

LaSDI-IT is motivated by a failure mode of standard autoencoder-based reduced-order models on interface-dominated problems. The stated difficulty is **spectral bias**: neural networks learn smooth, low-frequency content more easily than high-frequency or discontinuous structures. In shock-induced pore collapse, the state field is not smooth across the pore boundary, because the pore is represented by a zero-valued region while the surrounding high explosive material has nonzero temperature. Under those conditions, a conventional autoencoder can blur the pore boundary, generate nonphysical values in the void, mispredict interface motion, and degrade downstream quantities of interest [2507.10647].

The central design decision in LaSDI-IT is therefore to separate reconstruction of the smooth physical field from localization of the interface. This places the method within the broader LaSDI family of latent-space reduced-order models, but with a task-specific emphasis on discontinuity-rich systems rather than on generic parametric ROM construction. That emphasis is consistent with the broader LaSDI literature, which treats compression, latent dynamics identification, and parameter-space interpolation as modular components that can be altered for specific modeling constraints [2403.10748].

## 2. Interface-aware autoencoder architecture

The essential innovation is a revised autoencoder that jointly reconstructs the physical field \(T(x,t;p)\) and an indicator function \(\phi(x,t;p)\) describing the material region. In the pore-collapse application, the indicator is defined as
\[
\phi(x;p)=\mathbf{1}_{T>0}(x;p),
\]
so that \(\phi=1\) in the active HE region and \(\phi=0\) in the pore or void region. The encoder maps the temperature field to a latent variable \(z\in\mathbb{R}^{N_z}\), and the decoder outputs both a temperature-like field and an indicator prediction,
\[
(\tilde{T}',\phi_{\text{pred}})=De(z;\theta_{de}), \qquad z=En(T;\theta_{en}).
\]
The final reconstructed temperature is then obtained by masking:
\[
\tilde{T}_{\text{pred}}(x;p)=\tilde{T}'(x;p)\,\mathbf{1}_{\phi_{\text{pred}}(x;p)>0.5}.
\]
Accordingly, the decoder learns both field values and support of the field, rather than inferring the interface indirectly from smeared temperature values [2507.10647].

The autoencoder loss is replaced by a two-term objective,
\[
J_{AE}=J_{val}+J_{mask},
\]
where
\[
J_{val}=\frac{1}{N_pN_tN_x}\sum_{i=1}^{N_p}\sum_{k=1}^{N_t}\left\|T_k(p_i)-\tilde{T}_{k,\text{pred}}(p_i)\right\|^2,
\]
and
\[
J_{mask}=\frac{1}{N_pN_tN_x}\sum_{i=1}^{N_p}\sum_{k=1}^{N_t}\left\|\phi_k(p_i)-\phi_{k,\text{pred}}(p_i)\right\|^2.
\]
The decoder’s final output layer has width \(2N_x\), and a sigmoid activation is applied to the indicator branch so that \(\phi_{\text{pred}}\in[0,1]^{N_x}\). The reported interpretation is that the mask loss behaves like a gridwise classification task, which is easier than forcing the network to regress exact zeros inside the pore. This is presented as the reason the revised autoencoder converges better than a vanilla autoencoder on the pore-boundary reconstruction problem [2507.10647].

## 3. Masked preprocessing and latent dynamics identification

LaSDI-IT also revises the preprocessing step. Standard LaSDI scales the state using a global temporal mean and standard deviation, but pore values are treated here as physically meaningless placeholders for the indicator. LaSDI-IT therefore excludes pore values when computing statistics. It defines the active-sample count at each grid point,
\[
N_{\text{active}}(x_g)=\frac{1}{N_pN_t}\sum_{i=1}^{N_p}\sum_{k=1}^{N_t}\phi_k(x_g;p_i),
\]
and computes mean and standard deviation only over active values:
\[
\overline{T}(x_g)=\frac{1}{N_{\text{active}}(x_g)}\frac{1}{N_pN_t}\sum_{i=1}^{N_p}\sum_{k=1}^{N_t}T_k(x_g;p_i)\phi_k(x_g;p_i),
\]
\[
std[T](x_g)=\left\{\frac{1}{N_{\text{active}}(x_g)}\frac{1}{N_pN_t}\sum_{i=1}^{N_p}\sum_{k=1}^{N_t}\big(T_k(x_g;p_i)\phi_k(x_g;p_i)-\overline{T}(x_g)\big)^2\right\}^{1/2}.
\]
This masked scaling prevents the void region from contaminating the statistics and focuses the reduced representation on the active material [2507.10647].

After encoding, LaSDI-IT retains the LaSDI idea that the dynamics can be modeled in latent space by an ODE. For a field time history
\[
T(p)=\begin{pmatrix}T_0(p)&T_1(p)&\cdots&T_{N_t}(p)\end{pmatrix}\in\mathbb{R}^{N_x\times (N_t+1)},
\]
the encoder produces latent snapshots
\[
Z\equiv En(T)=\begin{pmatrix}z_1&z_2&\cdots&z_{N_t}\end{pmatrix}\in\mathbb{R}^{N_z\times N_t}.
\]
The latent dynamics are written as
\[
\dot z=\mathcal{L}[z;p].
\]
Although SINDy can support general nonlinear libraries, the reported pore-collapse implementation uses a **linear latent dynamics model**:
\[
\dot Z\approx (Z)\,\Xi(p), \qquad (Z)=\begin{pmatrix}\mathbf{1}&Z^\top\end{pmatrix}\in\mathbb{R}^{N_t\times (N_z+1)}.
\]
The coefficient matrix is fit by least squares,
\[
\Xi(p)=\frac{1}{\Delta t}(Z(p))^\dagger D_t Z(p)^\top,
\]
and the latent-dynamics residual in training is
\[
J_{LD}=\frac{1}{N_p}\sum_{i=1}^{N_p}\left\|\frac{1}{\Delta t}D_tZ(p_i)^\top-(Z(p_i))\,\Xi(p_i)\right\|^2.
\]
The total loss is
\[
J=J_{AE}+\beta_1J_{LD}+\beta_2\|\Xi\|_2^2.
\]
For the pore-collapse results, \(\beta_2=0\), and \(\beta_1\) is chosen on the order of \(\Delta t^2\). The paper gives two reasons for the linear latent dynamics choice: fewer coefficients, hence less overfitting in a data-scarce regime, and the assumption that the physical evolution after shock impact is largely one-directional and approaches a quasi-steady state, so nonlinear complexity is captured primarily by the autoencoder manifold [2507.10647].

## 4. Gaussian-process interpolation and greedy training-point selection

LaSDI-IT is parametric. In the pore-collapse study the physical parameter is
\[
p=(p_1,p_2),
\]
with \(p_1\) the pore orientation angle and \(p_2\) the pore major-axis length. For each latent coefficient entry \(\Xi_{ij}(p)\), the method uses Gaussian-process interpolation:
\[
\Xi_{ij}(p)\sim GP\!\left[\overline{\Xi}_{ij}(p),K(p,p')\right]+\epsilon,
\]
with zero mean and RBF kernel
\[
K(p,p';\theta_{GP})=\gamma\exp\!\left(-\frac{\|p-p'\|^2}{2\lambda^2}\right),
\]
where \(\theta_{GP}=\{\gamma,\lambda\}\). The hyperparameters are chosen by maximizing the marginal likelihood, with \(\sigma\) taken very small \((10^{-6})\). For an unseen parameter \(p^*\), the GP posterior mean and variance are
\[
\tilde{\Xi}_{ij}(p^*)=K(p^*,P)\left(K(P,P)+\sigma^2I\right)^{-1}D_{ij},
\]
\[
\mathrm{Var}(\Xi_{ij}(p^*))=K(p^*,p^*)-K(p^*,P)\left(K(P,P)+\sigma^2I\right)^{-1}K(P,p^*).
\]
The latent ODE is then integrated from the encoded initial condition
\[
z_0(p^*)=En(T_0(p^*)),
\]
and decoded back to the physical field [2507.10647].

This interpolation strategy directly reflects the GPLaSDI branch of the LaSDI family. GPLaSDI replaces deterministic interpolation of latent ODE coefficients with Gaussian-process regression, producing a predictive mean and predictive variance and enabling both uncertainty quantification and greedy adaptive sampling without requiring the PDE residual [2308.05882]. Within the broader LaSDI review literature, GPLaSDI is positioned as the non-intrusive branch of the framework, in contrast to residual-based intrusive variants such as gLaSDI [2403.10748].

LaSDI-IT adopts a GP-based greedy sampling rule. For candidate parameters \(\{p_1^*,\ldots,p_{N_p^*}^*\}\), uncertainty in the coefficients is propagated forward by Monte Carlo sampling. If \(T_{gp}\) denotes the tensor of predicted fields from \(N_s\) coefficient samples, the solution uncertainty is defined as
\[
std_{gp}[T_{gp}](p_i^*)=\max_{t_j}\frac{1}{N_x}\sum_{k=1}^{N_x}\frac{1}{N_s}\sum_{s=1}^{N_s}\left(T_{gp}(p_i^*,t_j,x_k,s)-\overline{T}_{gp}(p_i^*,t_j,x_k)\right)^2,
\]
with
\[
\overline{T}_{gp}(p_i^*,t_j,x_k)=\frac{1}{N_s}\sum_{s=1}^{N_s}T_{gp}(p_i^*,t_j,x_k,s).
\]
The next training point is selected by
\[
p_{N_p+1}=\arg\max_i\, std_{gp}[T_{gp}](p_i^*).
\]
This rule targets the point of maximal predicted uncertainty and is intended to reduce the number of expensive full-order simulations needed for retraining [2507.10647].

## 5. Shock-induced pore collapse benchmark

The reported application is shock-induced pore collapse in a high explosive. The setup consists of a \(3\,\mu\text{m}\) thick aluminum flyer impacting the HE, generating a shock pressure of about \(11.5\,\text{GPa}\). The simulation runs for \(3\,\text{ns}\) on a \(15\,\mu\text{m}\times15\,\mu\text{m}\) computational domain with grid spacing \(30\,\text{nm}\), using 300 time steps with \(\Delta t=0.01\,\text{ns}\). The surrogate is trained on a focused subdomain
\[
D=[0,5\,\mu\text{m}]\times[0,2\,\mu\text{m}],
\]
and the initial \(1.15\,\text{ns}\) are excluded because they mainly represent shock advection before pore interaction. The parameter space is
\[
p=(p_1,p_2)\in[0^\circ,20^\circ]\times[1\,\mu\text{m},1.2\,\mu\text{m}],
\]
where \(p_1\) is the pore orientation angle and \(p_2\) is the pore major-axis length; the pore minor axis is fixed at \(1\,\mu\text{m}\) [2507.10647].

Three quantities of interest are tracked:
\[
A_{hotspot}(t_k;p)=\int_D \mathbf{1}_{T_k(x;p)>800}\,dx^2,
\]
\[
A_{pore}(t_k;p)=\int_D \mathbf{1}_{T_k(x;p)=0}\,dx^2,
\]
\[
T_{k,\max}(p)=\max_x T_k(x;p).
\]
The hot spot threshold is \(800\,\text{K}\), stated to be above bulk heating but below the localized peak temperatures around the pore. The revised autoencoder is reported to reconstruct the pore boundary sharply, remove spurious nonzero pore values, and reduce the mask loss by about four orders of magnitude. With greedy GP sampling, LaSDI-IT reaches relative prediction error below \(\sim 9\%\) across the parameter space, with a worst-case example around \(9.3\%\), and yields performance comparable to a uniformly sampled training set of 36 points using only half the training data. The surrogate is also reported to reproduce pore area evolution, hot spot growth, timing of the temperature peak, and overall qualitative interface motion; the main residual error is a slight underestimation of the sharpest temperature peak, attributed to smoothing in latent space [2507.10647].

Large speedups are also reported, though the numerical summaries differ across the paper’s abstract and detailed description. The abstract states that the latent dynamics prediction was 106 times faster than the conventional high-fidelity simulation. The detailed summary reports that prediction for one parameter point takes about \(0.033\,\text{s}\), compared with about 10 minutes for a single high-fidelity ALE3D simulation on 112 CPU processors, corresponding to a speedup of roughly
\[
2.0\times 10^6
\]
in prediction time versus full simulation [2507.10647].

## 6. Position within the LaSDI ecosystem, limitations, and applicability

LaSDI-IT belongs to a family of latent-space ROMs whose common structure is compression, latent-dynamics identification, and reconstruction. Its distinguishing feature is that interface location becomes an explicit modeling target rather than a by-product of field regression. In that sense, it addresses a different failure mode from other LaSDI variants. WLaSDI alters the dynamics-identification step through weak-form regression to improve robustness to noise [2311.12880]. Rollout-LaSDI targets long-horizon accuracy by replacing purely local latent-dynamics objectives with a rollout loss [2509.08191]. mLaSDI addresses the reconstruction-versus-dynamics tradeoff through multi-stage residual correction [2506.09207]. GNN-LaSDI adapts the LaSDI concept to graph autoencoders for sharp-gradient systems on unstructured meshes [2606.23834]. LaSDI-IT is therefore best understood not as a replacement for the LaSDI framework, but as a specialized extension for moving-interface problems.

The reported limitations are specific. The latent dynamics are restricted to a linear ODE in the pore-collapse study; this is said to work well there, but may be insufficient for more strongly nonlinear dynamics. The indicator-based formulation assumes that the interface can be represented well by a binary mask, so more complex diffuse interfaces may require a different treatment. Peak localized temperatures can still be slightly smoothed, the study uses only two geometric parameters, and the present application involves a single pore rather than multiple interacting pores or more complex topologies [2507.10647].

The broader applicability stated for the method includes multiphase flows, fracture and crack propagation, phase change and solidification, detonation and reactive fronts, and shock/interface interactions in materials. This suggests that LaSDI-IT is most natural when a meaningful indicator or phase label can be defined and the dominant discontinuity can be represented as a moving interface. Under that condition, the combination of interface-aware reconstruction, latent ODE evolution, Gaussian-process interpolation, and greedy sampling provides a data-efficient reduced-order modeling strategy for discontinuity-rich computational physics [2507.10647].

Source: https://www.emergentmind.com/topics/lasdi-it