APRIL: Auxiliary Physically-Redundant Information in Loss - A physics-informed framework for parameter estimation with a gravitational-wave case study

TL;DR

PINNs struggle to scale to large datasets of realizations sharing the same physics. APRIL adds auxiliary physics-informed loss terms that exploit exact output redundancies, preserving the true minimum while reshaping the loss surface toward physically consistent solutions. In a gravitational-wave parameter-estimation case study using PN relations among $\mathcal{M}$, $M_{\mathrm{tot}}$, and $\eta$, APRIL yields up to an order-of-magnitude improvement in test accuracy, especially for the hard-to-learn $\eta$. This work demonstrates scalable, physically constrained learning for multi-system datasets and lays groundwork for integrating APRIL into future GW analyses with realistic noise and broader parameter ranges.

Abstract

Physics-Informed Neural Networks (PINNs) embed the partial differential equations (PDEs) governing the system under study directly into the training of Neural Networks, ensuring solutions that respect physical laws. While effective for single-system problems, standard PINNs scale poorly to datasets containing many realizations of the same underlying physics with varying parameters. To address this limitation, we present a complementary approach by including auxiliary physically-redundant information in loss (APRIL), i.e. augment the standard supervised output-target loss with auxiliary terms which exploit exact physical redundancy relations among outputs. We mathematically demonstrate that these terms preserve the true physical minimum while reshaping the loss landscape, improving convergence toward physically consistent solutions. As a proof-of-concept, we benchmark APRIL on a fully-connected neural network for gravitational wave (GW) parameter estimation (PE). We use simulated, noise-free compact binary coalescence (CBC) signals, focusing on inspiral-frequency waveforms to recover the chirp mass $\mathcal{M}$, the total mass $M_\mathrm{tot}$, and symmetric mass ratio $η$ of the binary. In this controlled setting, we show that APRIL achieves up to an order-of-magnitude improvement in test accuracy, especially for parameters that are otherwise difficult to learn. This method provides physically consistent learning for large multi-system datasets and is well suited for future GW analyses involving realistic noise and broader parameter ranges.

Figures (8)

• Figure 1: An example of $f(t_k)$ from the 1.5PN CBC GW event, corresponding to $m_1\simeq78.1\,M_\odot$, $m_2\simeq12.6\,M_\odot$ ($\mathcal{M}\simeq25.4$ M$_\odot$, $M_\textrm{tot}\simeq90.7$ M$_\odot$, and $\eta\simeq0.12$).
• Figure 2: Training, validation and test datasets for $\mathcal{M}$, $M_\mathrm{tot}$ and $\eta$. Training and validation datasets are generated by sampling $M_\mathrm{tot}$ and $\eta$ from a uniform distribution. The test dataset is obtained by sampling $m_1$ and $q$ from the mass distribution inferred by LVK collaboration from the GWTC-4 catalog theligoscientificcollaboration2025gwtc40updatinggravitationalwavetransienttheligoscientificcollaboration2025gwtc40populationpropertiesmerging, as described in Sec. \ref{['sec:test_dataset']}.
• Figure 3: Algorithm training flow. The frequency array $\left\{f_k\right\}_k^K$ is given as input to the FCNN architecture, to give $\left\{\mathcal{M},M_\mathrm{tot},\eta\right\}_\theta$ outputs. The outputs are then combined in different algebraic quantities that will be substituted in the different loss terms. The total loss is then computed as the sum of all terms. Its value and the gradient meta-data permits to update the NN parameter $\theta$ for a new epoch.
• Figure 4: RL1 results (median and 68% CI) for all the runs. Left panels show the study of the mass parameters individually, upper right panel shows the same study for the sum of the three, while bottom right panel shows the needed epochs to converge. The shape and the color of the markers determines the training dataset and the batch sizes. When APRIL losses are absent the RL1 result for the common sum is worse by a factor 10. Furthermore, the higher $D$ and the lower $B$, the better the results. Most of the APRIL impact is related to the $\eta$ term; see the text for discussion.
• Figure 5: Test output relative errors on mass components comparing the runs for $\left\{D,B,\text{seed}\right\}=\left\{5\times10^3,16,1\right\}$. The results for the two runs with active APRIL are similar to $\mathcal{L}_\mathrm{t}$ for $\mathcal{M}$ and $M_\mathrm{tot}$, but are more accurate for $\eta$, spanning only 3% in relative error instead of 10%.