Physics-Informed ML

Generalizing neural surrogate models across different PDE parameters remains difficult because changes in PDE coefficients often make learning harder and optimization less stable. The problem becomes even more severe when the model must also predict beyond the training time range. Existing methods usually cannot handle parameter generalization and temporal extrapolation at the same time. Standard parameterized models treat time as just another input and therefore fail to capture intrinsic dynamics, while recent continuous-time latent methods often rely on expensive test-time auto-decoding for each instance, which is inefficient and can disrupt continuity across the parameterized solution space. To address this, we propose Disentangled Latent Dynamics Manifold Fusion (DLDMF), a physics-informed framework that explicitly separates space, time, and parameters. Instead of unstable auto-decoding, DLDMF maps PDE parameters directly to a continuous latent embedding through a feed-forward network. This embedding initializes and conditions a latent state whose evolution is governed by a parameter-conditioned Neural ODE. We further introduce a dynamic manifold fusion mechanism that uses a shared decoder to combine spatial coordinates, parameter embeddings, and time-evolving latent states to reconstruct the corresponding spatiotemporal solution. By modeling prediction as latent dynamic evolution rather than static coordinate fitting, DLDMF reduces interference between parameter variation and temporal evolution while preserving a smooth and coherent solution manifold. As a result, it performs well on unseen parameter settings and in long-term temporal extrapolation. Experiments on several benchmark problems show that DLDMF consistently outperforms state-of-the-art baselines in accuracy, parameter generalization, and extrapolation robustness.

We establish empirical scaling laws for Single-Layer Physics-Informed Neural Networks on canonical nonlinear PDEs. We identify a dual optimization failure: (i) a baseline pathology, where the solution error fails to decrease with network width, even at fixed nonlinearity, falling short of theoretical approximation bounds, and (ii) a compounding pathology, where this failure is exacerbated by nonlinearity. We provide quantitative evidence that a simple separable power law is insufficient, and that the scaling behavior is governed by a more complex, non-separable relationship. This failure is consistent with the concept of spectral bias, where networks struggle to learn the high-frequency solution components that intensify with nonlinearity. We show that optimization, not approximation capacity, is the primary bottleneck, and propose a methodology to empirically measure these complex scaling effects.

Vision-based fall analysis has advanced rapidly, but a key bottleneck remains: visually similarmotions can correspond to very different physical outcomes because small differences in contactmechanics and protective responses are hard to infer from appearance alone. Most existingapproaches handle this by supervised injury prediction, which depends on reliable injury labels.In practice, such labels are difficult to obtain: video evidence is often ambiguous (occlusion,viewpoint limits), and true injury events are rare and cannot be safely staged, leading to noisysupervision. We address this problem with PHARL (PHysics-aware Alignment RepresentationLearning), which learns physically meaningful fall representations without requiring clinicaloutcome labels. PHARL regularizes motion embeddings with two complementary constraints:(1) trajectory-level temporal consistency for stable representation learning, and (2) multi-classphysics alignment, where simulation-derived contact outcomes shape embedding geometry. Bypairing video windows with temporally aligned simulation descriptors, PHARL captures localimpact-relevant dynamics while keeping inference purely feed-forward. Experiments on fourpublic datasets show that PHARL consistently improves risk-aligned representation quality overvisual-only baselines while maintaining strong fall-detection performance. Notably, PHARL alsoexhibits zero-shot ordinality: an interpretable severity structure (Head > Trunk > Supported)emerges without explicit ordinal supervision.

Physics-Informed Neural Networks (PINNs) have shown promise in solving incompressible Navier-Stokes equations, yet existing approaches are predominantly designed for single-flow settings. When extended to multi-flow scenarios, these methods face three key challenges: (1) difficulty in simultaneously capturing both shared physical principles and flow-specific characteristics, (2) susceptibility to inter-task negative transfer that degrades prediction accuracy, and (3) unstable training dynamics caused by disparate loss magnitudes across heterogeneous flow regimes. To address these limitations, we propose UniPINN, a unified multi-flow PINN framework that integrates three complementary components: a shared-specialized architecture that disentangles universal physical laws from flow-specific features, a cross-flow attention mechanism that selectively reinforces relevant patterns while suppressing task-irrelevant interference, and a dynamic weight allocation strategy that adaptively balances loss contributions to stabilize multi-objective optimization. Extensive experiments on three canonical flows demonstrate that UniPINN effectively unifies multi-flow learning, achieving superior prediction accuracy and balanced performance across heterogeneous regimes while successfully mitigating negative transfer. The source code of this paper will be released on https://github.com/Event-AHU/OpenFusion

Accurately upscaling terrestrial carbon fluxes is central to estimating the global carbon budget, yet remains challenging due to the sparse and regionally biased distribution of ground measurements. Existing data-driven upscaling products often fail to generalize beyond observed domains, leading to systematic regional biases and high predictive uncertainty. We introduce Task-Aware Modulation with Representation Learning (TAM-RL), a framework that couples spatio-temporal representation learning with knowledge-guided encoder-decoder architecture and loss function derived from the carbon balance equation. Across 150+ flux tower sites representing diverse biomes and climate regimes, TAM-RL improves predictive performance relative to existing state-of-the-art datasets, reducing RMSE by 8-9.6% and increasing explained variance (R2) from 19.4% to 43.8%, depending on the target flux. These results demonstrate that integrating physically grounded constraints with adaptive representation learning can substantially enhance the robustness and transferability of global carbon flux estimates.

(short version abstract, full in article)High-fidelity flow field reconstruction is important in fluid dynamics, but it is challenged by sparse and spatiotemporally incomplete sensor measurements, as well as failures of pre-deployed measurement points that can invalidate pre-trained reconstruction models. Physics-informed neural networks (PINNs) alleviate dependence on large labeled datasets by incorporating governing physics, yet sensor placement optimization, a key factor in reconstruction accuracy and robustness, remains underexplored. In this study, we propose a PINN with Voronoi-enhanced Sensor Optimization (VSOPINN). VSOPINN enables differentiable soft Voronoi construction for sparse sensor data rasterization, end-to-end fusion of centroidal Voronoi tessellation (CVT) with PINNs for adaptive sensor placement, and unified layout optimization for multi-condition flow reconstruction through a shared encoder-multi-decoder architecture. We validate VSOPINN on three representative problems: lid-driven cavity flow, vascular flow, and annular rotating flow. Results show that VSOPINN significantly improves reconstruction accuracy across different Reynolds numbers, adaptively learns effective sensor layouts, and remains robust under partial sensor failure. The study clarifies the intrinsic relationship between sensor placement and reconstruction precision in PINN-based flow field reconstruction.

Physics-informed neural networks (PINNs) have shown promise in addressing the ill-posed deconvolution problem in computed tomography perfusion (CTP) imaging for acute ischemic stroke assessment. However, existing PINN-based approaches remain deterministic and do not quantify uncertainty associated with violations of physics constraints, limiting reliability assessment. We propose Evidential Perfusion Physics-Informed Neural Networks (EPPINN), a framework that integrates evidential deep learning with physics-informed modeling to enable uncertainty-aware perfusion parameter estimation. EPPINN models arterial input, tissue concentration, and perfusion parameters using coordinate-based networks, and places a Normal--Inverse--Gamma distribution over the physics residual to characterize voxel-wise aleatoric and epistemic uncertainty in physics consistency without requiring Bayesian sampling or ensemble inference. The framework further incorporates physiologically constrained parameterization and stabilization strategies to promote robust per-case optimization. We evaluate EPPINN on digital phantom data, the ISLES 2018 benchmark, and a clinical cohort. On the evaluated datasets, EPPINN achieves lower normalized mean absolute error than classical deconvolution and PINN baselines, particularly under sparse temporal sampling and low signal-to-noise conditions, while providing conservative uncertainty estimates with high empirical coverage. On clinical data, EPPINN attains the highest voxel-level and case-level infarct-core detection sensitivity. These results suggest that evidential physics-informed learning can improve both accuracy and reliability of CTP analysis for time-critical stroke assessment. Source code is available at https://github.com/jhlee0619/EPPINN.

Scientific machine learning (SciML) is increasingly applied to in-field processing, controlling, and monitoring; however, wide-area sensing, real-time demands, and strict energy and reliability constraints make centralized SciML implementation impractical. Most SciML models assume raw data aggregation at a central node, incurring prohibitively high communication latency and energy costs; yet, distributing models developed for general-purpose ML often breaks essential physical principles, resulting in degraded performance. To address these challenges, we introduce EPIC, a hardware- and physics-co-guided distributed SciML framework, using full-waveform inversion (FWI) as a representative task. EPIC performs lightweight local encoding on end devices and physics-aware decoding at a central node. By transmitting compact latent features rather than high-volume raw data and by using cross-attention to capture inter-receiver wavefield coupling, EPIC significantly reduces communication cost while preserving physical fidelity. Evaluated on a distributed testbed with five end devices and one central node, and across 10 datasets from OpenFWI, EPIC reduces latency by 8.9$\times$ and communication energy by 33.8$\times$, while even improving reconstruction fidelity on 8 out of 10 datasets.

While Physics-Informed Neural Networks (PINNs) offer a mesh-free approach to solving fluid-flow PDEs, standard point-wise residual minimization suffers from convergence pathologies in topologically complex domains like Triply Periodic Minimal Surfaces (TPMS). The locality bias of point-wise constraints fails to propagate global information through tortuous channels, causing unstable gradients and conservation violations. To address this, we propose the Multi-scale Weak-form PINN (MUSA-PINN), which reformulates Navier-Stokes equation constraints as integral conservation laws over hierarchical spherical control volumes. We enforce continuity and momentum conservation via flux-balance residuals on control surfaces. Our method utilizes a three-scale subdomain strategy-comprising large volumes for long-range coupling, skeleton-aware meso-scale volumes aligned with transport pathways, and small volumes for local refinement-alongside a two-stage training schedule prioritizing continuity. Experiments on steady incompressible flow in TPMS geometries show MUSA-PINN outperforms state-of-the-art baselines, reducing relative errors by up to 93% and preserving mass conservation.

Reliable real-time estimation of atmospheric turbulence intensity remains an open challenge for aircraft operating across diverse altitude bands, particularly over oceanic, polar, and data-sparse regions that lack operational nowcasting infrastructure. Classical spectral models encode climatological averages rather than the instantaneous atmospheric state, and generic ML regressors offer adaptivity but provide no guarantee that predictions respect fundamental scaling laws. This paper introduces the Physically-Structured Turbulence Network (PSTNet), a lightweight architecture that embeds physics directly into its structure. PSTNet couples four components: (i) a zero-parameter backbone derived from Monin-Obukhov theory, (ii) a regime-gated mixture of specialist sub-networks supervised by Richardson-number-derived soft targets, (iii) Feature-wise Linear Modulation layers conditioning hidden representations on local air-density ratio, and (iv) a Kolmogorov output layer enforcing inertial-subrange scaling as an architectural constraint. The entire model contains only 552 learnable parameters, requiring fewer than 2.5 kB of storage and executing in under 12s on a Cortex-M7 microcontroller. We validate PSTNet on 340 paired six-degree-of-freedom guidance simulations spanning three vehicle classes (Mach 2.8, 4.5, and 8.0) and six operational categories with real-time satellite weather ingestion. PSTNet achieves a mean miss-distance improvement of +2.8% with a 78% win rate and a statistically significant effect size. Our results demonstrate that encoding domain physics as architectural priors yields a more efficient and interpretable path to turbulence estimation accuracy than scaling model capacity, establishing PSTNet as a viable drop-in replacement for legacy look-up tables in resource-constrained, safety-critical on-board guidance systems.

Embeddings provide low-dimensional representations that organize complex function spaces and support generalization. They provide a geometric representation that supports efficient retrieval, comparison, and generalization. In this work we generalize the concept to Physics Informed Neural Networks. We present a method to construct solution embedding spaces of nonlinear partial differential equations using a multi-head setup, and extract non-degenerate information from them using principal component analysis (PCA). We test this method by applying it to viscous Burgers' equation, which is solved simultaneously for a family of initial conditions and values of the viscosity. A shared network body learns a latent embedding of the solution space, while linear heads map this embedding to individual realizations. By enforcing orthogonality constraints on the heads, we obtain a principal-component decomposition of the latent space that is robust to training degeneracies and admits a direct physical interpretation. The obtained components for Burgers' equation exhibit rapid saturation, indicating that a small number of latent modes captures the dominant features of the dynamics.

For robots to navigate safely and efficiently on soft, granular terrains, it is crucial to gather information about the terrain's mechanical properties, which directly affect locomotion performance. Recent research has developed robotic legs that can accurately sense ground reaction forces during locomotion. However, existing tests of granular property estimation often rely on specific foot trajectories, such as vertical penetration or horizontal shear, limiting their applicability during natural locomotion. To address this limitation, we introduce a physics-informed machine learning framework, Inverse Resistive Force Theory (I-RFT), which integrates the Granular Resistive Force Theory model with Gaussian Processes to infer terrain properties from proprioceptively measured contact forces under arbitrary gait trajectories. By embedding the granular force model within the learning process, I-RFT preserves physical consistency while enabling generalization across diverse motion primitives. Experimental results demonstrate that I-RFT accurately estimates terrain properties across multiple gait trajectories and toe shapes. Moreover, we show that the quantified uncertainty over the terrain resistance stress map could enable robots to optimize foot design and gait trajectories for efficient information gathering. This approach establishes a new foundation for data-efficient characterization of complex granular environments and opens new avenues for locomotion strategies that actively adapt gait for autonomous terrain exploration.

Partial differential equations (PDEs) govern physical phenomena across the full range of scientific scales, yet their computational solution remains one of the defining challenges of modern science. This critical review examines two mature but epistemologically distinct paradigms for PDE solution, classical numerical methods and machine learning approaches, through a unified evaluative framework organized around six fundamental computational challenges. Classical methods are assessed for their structure-preserving properties, rigorous convergence theory, and scalable solver design; their persistent limitations in high-dimensional and geometrically complex settings are characterized precisely. Machine learning approaches are introduced under a taxonomy organized by the degree to which physical knowledge is incorporated and subjected to the same critical evaluation applied to classical methods. Classical methods are deductive -- errors are bounded by quantities derivable from PDE structure and discretization parameters -- while machine learning methods are inductive -- accuracy depends on statistical proximity to the training distribution. This epistemological distinction is the primary criterion governing responsible method selection. We identify three genuine complementarities between the paradigms and develop principles for hybrid design, including a framework for the structure inheritance problem that addresses when classical guarantees propagate through hybrid couplings, and an error budget decomposition that separates discretization, neural approximation, and coupling contributions. We further assess emerging frontiers, including foundation models, differentiable programming, quantum algorithms, and exascale co-design, evaluating each against the structural constraints that determine whether current barriers are fundamental or contingent on engineering progress.

Subsurface ore detection is of paramount importance given the rising depletion of shallow mineral resources in recent years. It is crucial to explore approaches that go beyond the limitations of traditional geological exploration methods. Due to readily available surface readings, joint magnetic and gravitational inversion is a promising new method - given magnetic and gravitational data on a surface, jointly reconstructing the underlying densities that generate them. However, this is ill-posed and has non-unique solutions. Deterministic methods often require handcrafted priors and converge to a single solution and do not capture the distribution, which is often of interest. We introduce a novel framework that reframes 3D gravity and magnetic joint inversion as a rectified flow on the Noddyverse dataset, the largest physics-based dataset for inversion. We introduce a Ginzburg-Landau (GL) regularizer, a generalized version of the Ising model that aids in ore identification, enabling physics-aware training. We also propose a guidance methodology based on GL theory that can be used as a plug-and-play module with existing unconditional denoisers. Lastly, we also train and release a VAE for the 3D densities, which facilitates downstream work in the field.

Solving flow through porous media is a crucial step in the topology optimisation of cold plates, a key component in modern thermal management. Traditional computational fluid dynamics (CFD) methods, while accurate, are often prohibitively expensive for large and complex geometries. In contrast, data-driven surrogate models provide a computationally efficient alternative, enabling rapid and reliable predictions. In this study, we develop a machine-learning framework for predicting steady-state flow through porous media governed by the Navier-Stokes-Brinkman equations. We implement and compare three model architectures-convolutional autoencoder (AE), U-Net, and Fourier Neural Operator (FNO)-evaluating their predictive performance. To enhance physics consistency, we incorporate physics-informed loss functions. Our results demonstrate that FNO outperforms AE and U-Net, achieving a mean squared error (MSE) as low as 0.0017 while providing speedups of up to 1000 times compared to CFD. Additionally, the mesh-invariant property of FNO emphasizes its suitability for topology optimisation tasks, where varying mesh resolutions are required. This study highlights the potential of machine learning to accelerate fluid flow predictions in porous media, offering a scalable alternative to traditional numerical methods.

Physics-Informed Neural Networks (PINNs) enforce governing equations by penalizing PDE residuals at interior collocation points, but standard collocation strategies - uniform sampling and residual-based adaptive refinement - can oversample smooth regions, produce highly correlated point sets, and incur unnecessary training cost. We reinterpret collocation selection as a coreset construction problem: from a large candidate pool, select a fixed-size subset that is simultaneously informative (high expected impact on reducing PDE error) and diverse (low redundancy under a space-time similarity notion). We formulate this as a QUBO/BQM objective with linear terms encoding residual-based importance and quadratic terms discouraging redundant selections. To avoid the scalability issues of dense k-hot QUBOs, we propose a sparse graph-based BQM built on a kNN similarity graph and an efficient repair procedure that enforces an exact collocation budget. We further introduce hybrid coverage anchors to guarantee global PDE enforcement. We evaluate the method on the 1D time-dependent viscous Burgers equation with shock formation and report both accuracy and end-to-end time-to-accuracy, including a timing breakdown of selection overhead. Results demonstrate that sparse and hybrid formulations reduce selection overhead relative to dense QUBOs while matching or improving accuracy at fixed collocation budgets.

Modeling stiff partial differential equations (PDEs) with sharp gradients remains a significant challenge for scientific machine learning. While Physics-Informed Neural Networks (PINNs) struggle with spectral bias and slow training times, Physics-Informed Extreme Learning Machines (PIELMs) offer a rapid, closed-form linear solution but are fundamentally limited by physics-agnostic, random initialization. We introduce the Gaussian Mixture Model Adaptive PIELM (GMM-PIELM), a probabilistic framework that learns a probability density function representing the ``location of physics'' for adaptively sampling kernels of PIELMs. By employing a weighted Expectation-Maximization (EM) algorithm, GMM-PIELM autonomously concentrates radial basis function centers in regions of high numerical error, such as shock fronts and boundary layers. This approach dynamically improves the conditioning of the hidden layer without the expensive gradient-based optimization(of PINNs) or Bayesian search. We evaluate our methodology on 1D singularly perturbed convection-diffusion equations with diffusion coefficients $ν=10^{-4}$. Our method achieves $L_2$ errors up to $7$ orders of magnitude lower than baseline RBF-PIELMs, successfully resolving exponentially thin boundary layers while retaining the orders-of-magnitude speed advantage of the ELM architecture.

Modern Earth observation relies on satellites to capture detailed surface properties. Yet, many phenomena that affect humans and ecosystems unfold in the atmosphere close to the surface. Near-ground sensors provide accurate measurements of certain environmental characteristics, such as near-surface air temperature (NSAT). However, they remain sparse and unevenly distributed, limiting their ability to provide continuous spatial measurements. To bridge this gap, we introduce SPyCer, a semi-supervised physics-guided network that can leverage pixel information and physical modeling to guide the learning process through meaningful physical properties. It is designed for continuous estimation of NSAT by proxy using satellite imagery. SPyCer frames NSAT prediction as a pixel-wise vision problem, where each near-ground sensor is projected onto satellite image coordinates and positioned at the center of a local image patch. The corresponding sensor pixel is supervised using both observed NSAT and physics-based constraints, while surrounding pixels contribute through physics-guided regularization derived from the surface energy balance and advection-diffusion-reaction partial differential equations. To capture the physical influence of neighboring pixels, SPyCer employs a multi-head attention guided by land cover characteristics and modulated with Gaussian distance weighting. Experiments on real-world datasets demonstrate that SPyCer produces spatially coherent and physically consistent NSAT estimates, outperforming existing baselines in terms of accuracy, generalization, and alignment with underlying physical processes.

Physics-informed neural networks (PINNs) are a versatile tool in the burgeoning field of scientific machine learning for solving partial differential equations (PDEs). However, determining suitable training strategies for them is not obvious, with the result that they typically yield moderately accurate solutions. In this article, we propose a method for improving the accuracy of PINNs by coupling them with a post-processing step that seeks the best approximation in a function space associated with the network. We find that our method yields errors four to five orders of magnitude lower than those of the parent PINNs across architectures and dimensions. Moreover, we can reuse the basis functions for the linear space in more complex settings, such as time-dependent and nonlinear problems, allowing for transfer learning. Our approach also provides a residual-based metric that allows us to optimally choose the number of basis functions employed.

Foundation models in general promise to accelerate scientific computation by learning reusable representations across problem instances, yet constrained scientific systems, where predictions must satisfy physical laws and safety limits, pose unique challenges that stress conventional training paradigms. We derive design principles for constrained scientific foundation models through systematic investigation of AC optimal power flow (ACOPF), a representative optimization problem in power grid operations where power balance equations and operational constraints are non-negotiable. Through controlled experiments spanning architectures, training objectives, and system diversity, we extract three empirically grounded principles governing scientific foundation model design. These principles characterize three design trade-offs: learning physics-invariant representations while respecting system-specific constraints, optimizing accuracy while ensuring constraint satisfaction, and ensuring reliability in high-impact operating regimes. We present the LUMINA framework, including data processing and training pipelines to support reproducible research on physics-informed, feasibility-aware foundation models across scientific applications.

Cardiac arrhythmogenesis is governed by complex electromechanical interactions that are not directly observable in vivo, motivating the development of non-invasive computational approaches for reconstructing three-dimensional activation dynamics. We present a physics-informed neural network framework for recovering cardiac activation patterns, active tension propagation, deformation fields, and hydrostatic pressure from measurable deformation data in simplified left ventricular geometries. Our approach integrates nonlinear anisotropic constitutive modeling, heterogeneous fiber orientation, weak formulations of the governing mechanics, and finite-element-based loss functions to embed physical constraints directly into training. We demonstrate that the proposed framework accurately reconstructs spatiotemporal activation dynamics under varying levels of measurement noise and reduced spatial resolution, while preserving global propagation patterns and activation timing. By coupling mechanistic modeling with data-driven inference, this method establishes a pathway toward patient-specific, non-invasive reconstruction of cardiac activation, with potential applications in digital phenotyping and computational support for arrhythmia assessment.

The numerical simulation of convection-dominated transient transport phenomena poses significant computational challenges due to sharp gradients and propagating fronts across the spatiotemporal domain. Classical discretization methods often generate spurious oscillations, requiring advanced stabilization techniques. However, even stabilized finite element methods may require additional regularization to accurately resolve localized steep layers. On the other hand, standalone physics-informed neural networks (PINNs) struggle to capture sharp solution structures in convection-dominated regimes and typically require a large number of training epochs. This work presents a hybrid computational framework that extends the PINN-Augmented SUPG with Shock-Capturing (PASSC) methodology from steady to unsteady problems. The approach combines a semi-discrete stabilized finite element method with a PINN-based correction strategy for transient convection-diffusion-reaction equations. Stabilization is achieved using the Streamline-Upwind Petrov-Galerkin (SUPG) formulation augmented with a YZbeta shock-capturing operator. Rather than training over the entire space-time domain, the neural network is applied selectively near the terminal time, enhancing the finite element solution using the last K_s temporal snapshots while enforcing residual constraints from the governing equations and boundary conditions. The network incorporates residual blocks with random Fourier features and employs progressive training with adaptive loss weighting. Numerical experiments on five benchmark problems, including boundary and interior layers, traveling waves, and nonlinear Burgers dynamics, demonstrate significant accuracy improvements at the terminal time compared to standalone stabilized finite element solutions.

Physics-Informed Neural Networks (PINNs) have been recognized as a mesh-free alternative to solve partial differential equations where physics information is incorporated. However, in dealing with problems characterized by high stiffness or shock-dominated dynamics, traditional PINNs have been found to have limitations, including unbalanced training and inaccuracy in solution, even with small physics residuals. In this research, we seek to address these limitations using the viscous Burgers' equation with low viscosity and the Allen-Cahn equation as test problems. In addressing unbalanced training, we have developed a new adaptive loss balancing scheme using smoothed gradient norms to ensure satisfaction of initial and boundary conditions. Further, to address inaccuracy in the solution, we have developed an adaptive residual-based collocation scheme to improve the accuracy of solutions in the regions with high physics residuals. The proposed new approach significantly improves solution accuracy with consistent satisfaction of physics residuals. For instance, in the case of Burgers' equation, the relative L2 error is reduced by about 44 percent compared to traditional PINNs, while for the Allen-Cahn equation, the relative L2 error is reduced by approximately 70 percent. Additionally, we show the trustworthy solution comparison of the proposed method using a robust finite difference solver.

Physics-Informed Neural Networks (PINNs) incorporate physics into neural networks by embedding partial differential equations (PDEs) into their loss function. Despite their success in learning the underlying physics, PINN models remain difficult to train and interpret. In this work, a novel modeling approach is proposed, which relies on the use of Domain-aware Fourier Features (DaFFs) for the positional encoding of the input space. These features encapsulate all the domain-specific characteristics, such as the geometry and boundary conditions, and unlike Random Fourier Features (RFFs), eliminate the need for explicit boundary condition loss terms and loss balancing schemes, while simplifying the optimization process and reducing the computational cost associated with training. We further develop an LRP-based explainability framework tailored to PINNs, enabling the extraction of relevance attribution scores for the input space. It is demonstrated that PINN-DaFFs achieve orders-of-magnitude lower errors and allow faster convergence compared to vanilla PINNs and RFFs-based PINNs. Furthermore, LRP analysis reveals that the proposed leads to more physically consistent feature attributions, while PINN-RFFs and vanilla PINNs display more scattered and less physics-relevant patterns. These results demonstrate that DaFFs not only enhance PINNs' accuracy and efficiency but also improve interpretability, laying the ground for more robust and informative physics-informed learning.

Creating functional Digital Twins, simulatable 3D replicas of the real world, is a central challenge in computer vision. Current methods like NeRF produce visually rich but functionally incomplete twins. The key barrier is the lack of underlying material properties (e.g., permittivity, conductivity). Acquiring this information for every point in a scene via non-contact, non-invasive sensing is a primary goal, but it demands solving a notoriously ill-posed physical inversion problem. Standard remote signals, like images and radio frequencies (RF), deeply entangle the unknown geometry, ambient field, and target materials. We introduce NEMF, a novel framework for dense, non-invasive physical inversion designed to build functional digital twins. Our key insight is a systematic disentanglement strategy. NEMF leverages high-fidelity geometry from images as a powerful anchor, which first enables the resolution of the ambient field. By constraining both geometry and field using only non-invasive data, the original ill-posed problem transforms into a well-posed, physics-supervised learning task. This transformation unlocks our core inversion module: a decoder. Guided by ambient RF signals and a differentiable layer incorporating physical reflection models, it learns to explicitly output a continuous, spatially-varying field of the scene's underlying material parameters. We validate our framework on high-fidelity synthetic datasets. Experiments show our non-invasive inversion reconstructs these material maps with high accuracy, and the resulting functional twin enables high-fidelity physical simulation. This advance moves beyond passive visual replicas, enabling the creation of truly functional and simulatable models of the physical world.

Many physical and engineering systems require solving direct problems to predict behavior and inverse problems to determine unknown parameters from measurement. In this work, we study both aspects for systems governed by differential equations, contrasting well-established numerical methods with new AI-based techniques, specifically Physics-Informed Neural Networks (PINNs). We first analyze the logistic differential equation, using its closed-form solution to verify numerical schemes and validate PINN performance. We then address the Porous Medium Equation (PME), a nonlinear partial differential equation with no general closed-form solution, building strong solvers of the direct problem and testing techniques for parameter estimation in the inverse problem. Our results suggest that PINNs can closely estimate solutions at competitive computational cost, and thus propose an effective tool for solving both direct and inverse problems for complex systems.

Training neural PDE solvers is often bottlenecked by expensive data generation or unstable physics-informed neural network (PINN) involving challenging optimization landscapes due to higher-order derivatives. To tackle this issue, we propose an alternative approach using Monte Carlo approaches to estimate the solution to the PDE as a stochastic process for weak supervision during training. Leveraging the Walk-on-Spheres method, we introduce a learning scheme called Walk-on-Spheres Neural Operator (WoS-NO) which uses weak supervision from WoS to train any given neural operator. We propose to amortize the cost of Monte Carlo walks across the distribution of PDE instances using stochastic representations from the WoS algorithm to generate cheap, noisy, estimates of the PDE solution during training. This is formulated into a data-free physics-informed objective where a neural operator is trained to regress against these weak supervisions, allowing the operator to learn a generalized solution map for an entire family of PDEs. This strategy does not require expensive pre-computed datasets, avoids computing higher-order derivatives for loss functions that are memory-intensive and unstable, and demonstrates zero-shot generalization to novel PDE parameters and domains. Experiments show that for the same number of training steps, our method exhibits up to 8.75$\times$ improvement in $L_2$-error compared to standard physics-informed training schemes, up to 6.31$\times$ improvement in training speed, and reductions of up to 2.97$\times$ in GPU memory consumption. We present the code at https://github.com/neuraloperator/WoS-NO

A data-free finite-volume convolutional neural network (FV-CNN) is formulated for steady Euler flow around a cylinder at eight Mach numbers, 2--15. Shared-face fluxes define conservative spatial balances, while automatic differentiation supplies parameter gradients. The derivative mismatch in the earlier automatic-differentiation-based convolutional neural network (AD-CNN) is derived. No computational fluid dynamics (CFD) or experimental field targets enter training; reference-independent selection of coefficients, seeds, and checkpoints is not established. Billig-based upstream fixing conditions shock-location agreement.For eight adopted fields, density and pressure relative $L_2$ discrepancies from finite-resolution CFD are $5.43$--$18.69\%$ and $6.25$--$19.70\%$. Closed-band defects are smaller than AD-CNN in 32/32 self-centered and 22--31/32 CFD-centered comparisons over seven half-widths, 0.06--0.12 cylinder diameters. AD-CNN retains smaller enthalpy root-mean-square (RMS) discrepancies in 7/8 cases, shock-position RMS at Mach 11 and 15, and temperature and transverse-velocity discrepancies at Mach 3.Ten new Mach-15 initializations and adopted seed 42 yield density discrepancies of $12.04$--$43.70\%$ (mean $24.34\%$, sample standard deviation $11.78$ percentage points); the new ten alone give $25.04\%$ and $12.17$ points. Dispersion exceeds the historical adopted-field gap of $3.40$ points; six of eleven outcomes are below AD-CNN. Transition-width ratios of AD-CNN to FV-CNN remain $4.07$--$4.66$, and all 44 self-centered seed--component balances at half-width 0.10 cylinder diameters are smaller for FV-CNN. Local sharpness and this balance metric persist, while initialization-robust accuracy superiority is not established.Component interventions remain one-seed conditional responses; Mach 2 uses a separate budget-limited protocol.

Accurate weight estimation of commercial and industrial waste is important for efficient operations, yet image-based estimation remains difficult because similar-looking objects may have different densities, and the visible size changes with camera distance. Addressing this problem, we propose Multimodal Weight Predictor (MWP) framework that estimates waste weight by combining RGB images with physics-informed metadata, including object dimensions, camera distance, and camera height. We also introduce Waste-Weight-10K, a real-world dataset containing 10,421 synchronized image-metadata collected from logistics and recycling sites. The dataset covers 11 waste categories and a wide weight range from 3.5 to 3,450 kg. Our model uses a Vision Transformer for visual features and a dedicated metadata encoder for geometric and category information, combining them with Stacked Mutual Attention Fusion that allows visual and physical cues guide each other. This helps the model manage perspective effects and link objects to material properties. To ensure stable performance across the wide weight range, we train the model using Mean Squared Logarithmic Error. On the test set, the proposed method achieves 88.06 kg Mean Absolute Error (MAE), 6.39% Mean Absolute Percentage Error (MAPE), and an R2 coefficient of 0.9548. The model shows strong accuracy for light objects in the 0-100 kg range with 2.38 kg MAE and 3.1% MAPE, maintaining reliable performance for heavy waste in the 1000-2000 kg range with 11.1% MAPE. Finally, we incorporate a physically grounded explanation module using Shapley Additive Explanations (SHAP) and a large language model to provide clear, human-readable explanations for each prediction.

Inferring biophysical parameters and hidden state variables from partial and noisy observations is a fundamental challenge in computational neuroscience. This problem is particularly difficult for fast - slow spiking and bursting models, where strong nonlinearities, multiscale dynamics, and limited observational data often lead to severe sensitivity to initial parameter guesses and convergence failure in the methods replying on the traditional numerical forward solvers. In this work, we developed a physics-informed neural network (PINN) framework for the joint reconstruction of unobserved state variables and the estimation of unknown biophysical parameters in neuronal models. We demonstrate the effectiveness of the method on biophysical neuron models, including the Morris-Lecar model across multiple spiking and bursting regimes and a respiratory model neuron. The method requires only partial voltage observations over short observation windows and remains robust even when initialized with non-informative parameter guesses. These results suggest that PINN can deliver robust and accurate parameter inference and state reconstruction, providing a promising alternative for inverse problems in multiscale neuronal dynamics, where traditional techniques often struggle.