PILA: Physics-Informed Low Rank Augmentation for Interpretable Earth Observation

TL;DR

PILA tackles the challenge of inverting incomplete physical models in Earth Observation by augmenting forward physics with a learnable, low-rank residual. The method preserves physical interpretability while enabling flexible correction through a compact residual representation, controlled by rank and regularization. Across forest radiative transfer and volcanic GNSS deformations, PILA yields more plausible physical variables and robust reconstructions than the state-of-the-art HVAE, while also providing improved separation of biophysical factors and clear separation of transient versus seasonal signals. The work demonstrates PILA’s potential as a general framework for inversions with incomplete physics and highlights how rank, priors, and observability govern retrieval quality, with broad implications for EO and beyond.

Abstract

Physically meaningful representations are essential for Earth Observation (EO), yet existing physical models are often simplified and incomplete. This leads to discrepancies between simulation and observations that hinder reliable forward model inversion. Common approaches to EO inversion either ignored this incompleteness or relied on case-specific preprocessing. More recent methods use physics-informed autoencoders but depend on auxiliary variables that are difficult to interpret and multiple regularizers that are difficult to balance. We propose Physics-Informed Low-Rank Augmentation (PILA), a framework that augments incomplete physical models using a learnable low-rank residual to improve flexibility, while remaining close to the governing physics. We evaluate PILA on two EO inverse problems involving diverse physical processes: forest radiative transfer inversion from optical remote sensing; and volcanic deformation inversion from Global Navigation Satellite Systems (GNSS) displacement data. Across different domains, PILA yields more accurate and interpretable physical variables. For forest spectral inversion, it improves the separation of tree species and, compared to ground measurements, reduces prediction errors by 40-71\% relative to the state-of-the-art. For volcanic deformation, PILA's recovery of variables captures a major inflation event at the Akutan volcano in 2008, and estimates source depth, volume change, and displacement patterns that are consistent with prior studies that however required substantial additional preprocessing. Finally, we analyse the effects of model rank, observability, and physical priors, and suggest that PILA may offer an effective general pathway for inverting incomplete physical models even beyond the domain of Earth Observation. The code is available at https://github.com/yihshe/PILA.git.

Figures (29)

• Figure 1: We study inverse problems arising from Earth observations of disparate physical processes, to understand the planet, from the surface to the subsurface.Study cases include: the inversion of a forest radiative transfer model---a planet renderer---to estimate biophysical status; and the inversion of a volcanic deformation model---representative of a broad family of geophysical inverse problems---to infer subsurface geophysical activity.
• Figure 2: PILA inverts physical models of varying incompleteness using a residual of intrinsically low rank.Given an observation $X$, $\mathcal{E}_{\mathrm{R}}$ maps it to a high-dimensional feature $R$, which is then encoded by $\mathcal{E}_{\mathrm{phy}}$ and $\mathcal{E}_{\mathrm{aux}}$ into physical variables $Z_{\mathrm{phy}}$ and auxiliary variables $Z_{\mathrm{aux}}$. $Z_{\mathrm{phy}}$ is decoded by $\mathcal{F}$ to produce a physical reconstruction $X_{\mathcal{F}}$, which is refined by a low-rank residual $\Delta$ to yield the final reconstruction $X_{\mathcal{C}}$. The residual $\Delta$ is computed by a mapping $\mathcal{C}$ as the product of a scaling factor $s$, a coefficient matrix $\mathbf{A}\in\mathbb{R}^{n\times r}$, and a residual basis matrix $\mathbf{B}\in\mathbb{R}^{d\times r}$, with $\operatorname{rank}(\Delta)= r\ll d$. Here, $s$ and $\mathbf{B}$ are shared parameters applied to all samples, while $\mathbf{A}$ is obtained by linearly mapping the concatenation of $Z_{\mathrm{aux}}$ and the physical output $X_{\mathcal{F}}$, with a stop-gradient operation applied to $X_{\mathcal{F}}$ during backpropagation.
• Figure 3: HVAE contains multiple regularisation terms that can be difficult to tune in practice.Given an observation $X$, as in PILA, $\mathcal{E}_{\mathrm{R}}$ maps it to a high-dimensional feature $R$. A physical pathway predicts $Z_{\mathrm{phy}}$ and reconstructs $X_{\mathcal{F}}$, while an auxiliary pathway predicts $Z_{\mathrm{aux}}$ and augments $X_{\mathcal{F}}$ to produce the refined reconstruction $X_{\mathcal{C}}$. The architecture also includes an "unmixing" step to separate the modeled observation from the raw observation.: $R$ is passed to $\mathcal{E}_{\mathrm{unmix}}$ to obtain coefficients $\alpha_{\mathrm{unmix}}$, which are multiplied element-wise with $X$ to yield the unmixed observation $X_{\mathrm{unmix}}$. This is fed again to $\mathcal{E}_{\mathrm{R}}$ to obtain the unmixed feature $R_{\mathrm{unmix}}$, followed by $\mathcal{E}_{\mathrm{phy}}$ to produce physical variables $Z_{\mathrm{phy}}$. Then $Z_{\mathrm{phy}}$ is concatenated with $Z_{\mathrm{aux}}$ and passed through the decoder $\mathcal{D}_{\mathrm{aux}}$ to obtain the auxiliary feature $X_{\mathrm{aux}}$. Finally, $X_{\mathrm{aux}}$ is concatenated with $X_{\mathcal{F}}$ and mapped by a non-linear function $\mathcal{C}$ to produce $X_{\mathcal{C}}$. Several regularisation terms are added to enforce desired behavior.
• Figure 4: PILA (with $r=2$) yields more plausible distributions of physical variables.In contrast, variables recovered by HVAE frequently fall at boundary values. For example, $Z_{\mathrm{cm}}=0$ means no dry matter --- but there is foliage present.
• Figure 5: PILA achieves less perfect reconstruction of spectral bands ($r=2$).HVAE attains slightly higher reconstruction accuracy than PILA but, importantly, frequently recovers implausible values of physical variables (see \ref{['fig:rtm_inversion_austria_variables']}).