Reaction-Diffusion Systems

We present a global-in-time spectral Duhamel-Picard method with polynomial expansions for post-exposure bake (PEB) simulation. The coupled acid-quencher reaction-diffusion equations are reformulated as nonlinear Volterra integral equations using the Duhamel principle. Full trajectories are represented by finite polynomial expansions, with temporal coefficient fields determined over the bake interval by Anderson-accelerated Picard iteration, thereby avoiding sequential time marching. In 3-D PEB simulations, we employ Chebyshev and Legendre temporal bases while evaluating the Duhamel integrals using the corresponding Gaussian quadrature rules. We demonstrate that the proposed method simulates a large domain of $1200\times1200\times100$ voxels on the order of one minute. The representation is further incorporated into a physics-informed neural operator (PINO) with a mixed-transform variant of a factorized Fourier neural operator, which is trained without precomputed solution trajectories. Together, the integral formulation and its PINO extension enable accelerated, time-marching-free prediction of concentration fields at arbitrary times within the bake interval.

Generalized reaction-diffusion systems encompass diverse transport mechanisms and coupled reaction kinetics. A central question for neural PDE solvers is what should be learned so that a common interface can accommodate phase-field and degenerate transport, local reactions, and multispecies coupling. We propose the Neural Constitutive Laws--Mass-Compression-Transport (NCL-MCT) Solver, which learns PDE-specific constitutive responses while retaining temporal evolution in a shared MCT integrator. Transport is represented through mobility and thermodynamic driving force, and reaction through relative reaction rates. These constitutive responses depend on the current density rather than explicitly on the initial condition or elapsed time, motivating their reuse across different initial conditions and time horizons. The same interface supports velocity-data supervision and known-law supervision, neither of which requires time integration during training. When constitutive laws are known, supervision can be evaluated on independently sampled density fields, enabling trajectory-free constitutive learning without generating solution trajectories. Across seven systems, separately trained constitutive modules share the same interface and MCT integrator and achieve relative rollout $L^2$ errors of $10^{-4}$ to $10^{-2}$. Tests with unseen initial-condition families and an extended time horizon assess reuse beyond training conditions, while separate experiments demonstrate trajectory-free constitutive learning. These results support constitutive responses as an effective learning target for a shared neural PDE framework.

Learning surrogates for time-dependent partial differential equations often requires a new simulation corpus when the governing operator changes. We introduce RD-JEPA, a joint-embedding predictive architecture for self-supervised pretraining on reaction-diffusion trajectories. A single model is pretrained on five parameterized systems and then adapted to three held-out systems whose reaction operators and trajectories are excluded from pretraining. Using one, five, or ten complete trajectories from a held-out system, RD-JEPA achieves lower mean relative discrete $\ell^2$ field error and mean absolute spatial first-difference error than five supervised surrogate baselines, an independently trained control that removes the trajectory-dependent predictive latent pathway, and an architecture-matched model trained from scratch. Within the evaluated equations, output resolution, forecast horizons, and choices of adaptation trajectories, the results indicate that prediction of future-state representations can support data-efficient adaptation across related reaction-diffusion systems.

Brain tumor progression exhibits spatially heterogeneous growth, patient-specific treatment response, and complex interactions with surrounding anatomy, making accurate long-term prediction challenging. We propose an AI-augmented adaptive digital twin (DT) framework for brain tumor evolution prediction and treatment scheduling. The framework integrates an interpretable reaction--diffusion (RD) model, a 3D residual learning module for model-form correction, patient-specific DT updating during recursive rollout, and model predictive control (MPC) for constrained chemotherapy and radiotherapy scheduling. Experiments on 387 synthetic tumor trajectories with 120-step evolution show that the baseline RD model captures tumor location and overall temporal behavior but underestimates heterogeneous tumor burden during long-horizon prediction. Hybrid RD--residual modeling reduces masked voxel-wise mean squared error by 84.3% and increases Dice overlap by 43.5% relative to the RD baseline under dense simulated observations. Online DT updating further reduces mean squared error by 45.9% and improves Dice overlap by 9.6% compared with the non-updated hybrid model. In MPC-based scheduling simulations, the updated DT controller reduces final tumor burden by 22.4% relative to a fixed treatment schedule under the terminal-burden objective. Together, these results demonstrate a unified framework for patient-specific initialization, mechanistic modeling, adaptive learning, and constrained treatment optimization. Although validated using patient-data-informed synthetic trajectories rather than clinical longitudinal data, the proposed framework establishes a foundation for future translation to real-world adaptive treatment planning.

Gradient-based inversion of reaction-diffusion systems is typically approached via surrogate models or physics-informed neural networks (PINNs), while the most direct route, backpropagation through the PDE's structure itself, has largely been avoided. We pursue this direct route as a diagnostic probe, backpropagating a steady-state loss through unrolled Gray-Scott simulation to recover its parameters, with no surrogate or neural-network augmentation. Optimization fails to converge, and plotting the landscape directly locates the failure in its geometry -- flat plateaus with no gradient signal, bounded by sharp cliffs that align with bifurcation boundaries -- a structure that recurs across loss functions and is inherited however the gradients are routed to parameters. Reading this minimal setup as an ablation of PINN, we disentangle each component's role: with the neural network fixed, the residual loss is quadratic in the PDE parameters and yields a smooth landscape, so it alone already avoids the pathology, by implicitly encoding the full PDE dynamics across all initial conditions. The neural network, for its part, cannot repair an ill-posed parameter subspace, and so serves only to complete the observed data -- a division of labor not previously made explicit. These findings carry concrete design implications for PINN-type methods and a broader heuristic on when added dimensions actually help.

Video-language models are pivotal for tasks such as moment retrieval and highlight detection, yet they often struggle to capture the dynamic, non-linear interactions between temporal video sequences and textual semantics. Existing approaches, relying on static cross-attention or prompt-tuning mechanisms, fail to adaptively model the evolving relationships between modalities, leading to suboptimal alignment and limited generalization. Inspired by systems biology, we propose Reaction-Diffusion Multimodal Fusion (RDMF), a novel framework that reimagines video-language alignment as a reaction-diffusion (RD) process, drawing on the principles of pattern formation introduced by Alan Turing. In RDMF, video features diffuse across time to capture temporal context, while text-video interactions are modeled as non-linear reactions that amplify relevant features and suppress noise, forming emergent patterns akin to biological systems. Leveraging the Gray-Scott RD model, we design a computationally efficient fusion module that integrates video and text representations, supported by rigorous mathematical analysis of stability and convergence using Turing instability criteria. Our framework is theoretically grounded, employing advanced mathematical tools to ensure stable pattern formation, and is practically viable, incorporating standard components like pretrained encoders and DETR-style heads for moment retrieval and saliency prediction. RDMF represents a pioneering interdisciplinary approach, bridging systems biology and multimedia research to address the limitations of conventional multimodal fusion. Preliminary experiments demonstrate its potential to outperform existing methods in identifying salient video moments, offering a new paradigm for video-language tasks.

Neural operators provide a framework for learning solution operators of partial differential equations (PDEs), enabling efficient surrogate modeling for complex systems. While universal approximation results are now well understood, quantitative approximation analysis specific to nonlinear reaction-diffusion systems remains limited. In this paper, we study Laplacian-based neural operators applied to the solution mapping from initial conditions to time-dependent solutions of nonlinear reaction-diffusion systems. By exploiting the Laplacian spectral representation of the diffusion semigroup, we obtain polynomial complexity bounds on the network depth, width, and spectral rank, demonstrating that Laplacian eigenfunction-based neural operators can exploit the structure of the underlying PDE while avoiding the curse of parametric complexity encountered in generic operator learning. Numerical experiments support the theoretical findings and illustrate the benefit of Laplacian eigenfunction representations compared with Fourier-based spectral representations.

Physics-informed neural networks (PINNs) provide a powerful framework for learning governing equations of dynamical systems from data. Biologically-informed neural networks (BINNs) are a variant of PINNs that preserve the known differential operator structure (e.g., reaction-diffusion) while learning constitutive terms via trainable neural subnetworks, enforced through soft residual penalties. Existing BINN studies are limited to $1\mathrm{D}{+}t$ reaction-diffusion systems and focus on forward prediction, using the governing partial differential equation as a regulariser rather than an explicit identification target. Here, we extend BINNs to $2\mathrm{D}{+}t$ systems within a PINN framework that combines data preprocessing, BINN-based equation learning, and symbolic regression post-processing for closed-form equation discovery. We demonstrate the framework's real-world applicability by learning the governing equations of lung cancer cell population dynamics from time-lapse microscopy data, recovering $2\mathrm{D}{+}t$ reaction-diffusion models from experimental observations. The proposed framework is readily applicable to other spatio-temporal systems, providing a practical and interpretable tool for fast analytic equation discovery from data.

We investigate the data-driven discovery of constitutive closures in nonlinear reaction-diffusion systems with known governing PDE structures. Our objective is to robustly recover diffusion and reaction laws from spatiotemporal observations while avoiding the common pitfall where low residuals or short-horizon predictions are conflated with physical recovery. We propose a three-stage neural-symbolic framework: (1) learning numerical surrogates under physical constraints using a noise-robust weak-form-driven objective; (2) compressing these surrogates into restricted interpretable symbolic families (e.g., polynomial, rational, and saturation forms); and (3) validating the symbolic closures through explicit forward re-simulation on unseen initial conditions. Extensive numerical experiments reveal two distinct regimes. Under matched-library settings, weak polynomial baselines behave as correctly specified reference estimators, showing that neural surrogates do not uniformly outperform classical bases. Conversely, under function-class mismatch, neural surrogates provide necessary flexibility and can be compressed into compact symbolic laws with minimal rollout degradation. However, we identify a critical "bias inheritance" mechanism where symbolic compression does not automatically repair constitutive bias. Across various observation regimes, the true error of the symbolic closure closely tracks that of the neural surrogate, yielding a bias inheritance ratio near one. These findings demonstrate that the primary bottleneck in neural-symbolic modeling lies in the initial numerical inverse problem rather than the subsequent symbolic compression. We underscore that constitutive claims must be rigorously supported by forward validation rather than residual minimization alone.

World models learn to predict future states of an environment, enabling planning and mental simulation. Current approaches default to Transformer-based predictors operating in learned latent spaces. This comes at a cost: O(N^2) computation and no explicit spatial inductive bias. This paper asks a foundational question: is self-attention necessary for predictive world modeling, or can alternative computational substrates achieve comparable or superior results? I introduce FluidWorld, a proof-of-concept world model whose predictive dynamics are governed by partial differential equations (PDEs) of reaction-diffusion type. Instead of using a separate neural network predictor, the PDE integration itself produces the future state prediction. In a strictly parameter-matched three-way ablation on unconditional UCF-101 video prediction (64x64, ~800K parameters, identical encoder, decoder, losses, and data), FluidWorld is compared against both a Transformer baseline (self-attention) and a ConvLSTM baseline (convolutional recurrence). While all three models converge to comparable single-step prediction loss, FluidWorld achieves 2x lower reconstruction error, produces representations with 10-15% higher spatial structure preservation and 18-25% more effective dimensionality, and critically maintains coherent multi-step rollouts where both baselines degrade rapidly. All experiments were conducted on a single consumer-grade PC (Intel Core i5, NVIDIA RTX 4070 Ti), without any large-scale compute. These results establish that PDE-based dynamics, which natively provide O(N) spatial complexity, adaptive computation, and global spatial coherence through diffusion, are a viable and parameter-efficient alternative to both attention and convolutional recurrence for world modeling.

Bacterial cancer therapy exploits anaerobic bacteria's ability to target hypoxia tumor regions, yet the interactions among tumor growth, bacterial colonization, oxygen levels, immunosuppressive cytokines, and bacterial communication remain poorly quantified. We present a mathematical model of five coupled nonlinear reaction-diffusion equations in a two-dimensional tissue domain. We proved the global well-posedness of the model and identified its steady states to analyze stability. Furthermore, a physics-informed neural network (PINN) solves the system without a mesh and without requiring extensive data. It provides convergence guarantees by combining residual stability and Sobolev approximation error bounds. This results in an overall error rate of O(n^-2 ln^4(n) + N^-1/2), which depends on the network width n and the number of collocation points N. We conducted several numerical experiments, including predicting the tumor's response to therapy. We also performed a sensitivity analysis of certain parameters. The results suggest that long-term therapeutic efficacy may require the maintenance of hypoxia regions in the tumor, or using bacteria that tolerate oxygen better, may be necessary for long-lasting tumor control.

We propose an efficient and generalizable physics-informed neural network (PINN) framework for computing traveling wave solutions of $n$-dimensional reaction-diffusion equations with various reaction and diffusion coefficients. By applying a scaling transformation with the traveling wave form, the original problem is reduced to a one-dimensional scaled reaction-diffusion equation with unit reaction and diffusion coefficients. This reduction leads to the proposed framework, termed scaled TW-PINN, in which a single PINN solver trained on the scaled equation is reused for different coefficient choices and spatial dimensions. We also prove a universal approximation property of the proposed PINN solver for traveling wave solutions. Numerical experiments in one and two dimensions, together with a comparison to the existing wave-PINN method, demonstrate the accuracy, flexibility, and superior performance of scaled TW-PINN. Finally, we explore an extension of the framework to the Fisher's equation with general initial conditions.

Physics-guided sampling with diffusion model priors has shown promise for solving partial differential equation (PDE) governed problems, but applications to chemically meaningful reaction-transport systems remain limited. We apply diffusion-based guided sampling to gas-phase chemical reactions by training on solutions of the advection-reaction-diffusion (ARD) equation across varying parameters. The method generates physically consistent concentration fields and accurately predicts outlet concentrations, including at unseen parameter values, demonstrating the potential of diffusion models for inference in reactive transport.