Deep learning models trained on finite data lack a complete understanding of the physical world. On the other hand, physics-informed neural networks (PINNs) are infused with such knowledge through the incorporation of mathematically expressible laws of nature into their training loss function. By complying with physical laws, PINNs provide advantages over purely data-driven models in limited-data regimes. This feature has propelled them to the forefront of scientific machine learning, a domain characterized by scarce and costly data. However, the vision of accurate physics-informed learning comes with significant challenges. This review examines PINNs for the first time in terms of model optimization and generalization, shedding light on the need for new algorithmic advances to overcome issues pertaining to the training speed, precision, and generalizability of today's PINN models. Of particular interest are the gradient-free methods of neuroevolution for optimizing the uniquely complex loss landscapes arising in PINN training. Methods synergizing gradient descent and neuroevolution for discovering bespoke neural architectures and balancing multiple conflicting terms in physics-informed learning objectives are positioned as important avenues for future research. Yet another exciting track is to cast neuroevolution as a meta-learner of generalizable PINN models.

The potential of learned models for fundamental scientific research and discovery is drawing increasing attention worldwide. Physics-informed neural networks (PINNs), where the loss function directly embeds governing equations of scientific phenomena, is one of the key techniques at the forefront of recent advances. PINNs are typically trained using stochastic gradient descent methods, akin to their deep learning counterparts. However, analysis in this paper shows that PINNs' unique loss formulations lead to a high degree of complexity and ruggedness that may not be conducive for gradient descent. Unlike in standard deep learning, PINN training requires globally optimum parameter values that satisfy physical laws as closely as possible. Spurious local optimum, indicative of erroneous physics, must be avoided. Hence, neuroevolution algorithms, with their superior global search capacity, may be a better choice for PINNs relative to gradient descent methods. Here, we propose a set of five benchmark problems, with open-source codes, spanning diverse physical phenomena for novel neuroevolution algorithm development. Using this, we compare two neuroevolution algorithms against the commonly used stochastic gradient descent, and our baseline results support the claim that neuroevolution can surpass gradient descent, ensuring better physics compliance in the predicted outputs. %Furthermore, implementing neuroevolution with JAX leads to orders of magnitude speedup relative to standard implementations.

Physics-informed neural networks (PINNs) are at the forefront of scientific machine learning, making possible the creation of machine intelligence that is cognizant of physical laws and able to accurately simulate them. In this paper, the potential of discovering PINNs that generalize over an entire family of physics tasks is studied, for the first time, through a biological lens of the Baldwin effect. Drawing inspiration from the neurodevelopment of precocial species that have evolved to learn, predict and react quickly to their environment, we envision PINNs that are pre-wired with connection strengths inducing strong biases towards efficient learning of physics. To this end, evolutionary selection pressure (guided by proficiency over a family of tasks) is coupled with lifetime learning (to specialize on a smaller subset of those tasks) to produce PINNs that demonstrate fast and physics-compliant prediction capabilities across a range of empirically challenging problem instances. The Baldwinian approach achieves an order of magnitude improvement in prediction accuracy at a fraction of the computation cost compared to state-of-the-art results with PINNs meta-learned by gradient descent. This paper marks a leap forward in the meta-learning of PINNs as generalizable physics solvers.

We present a novel loss formulation for efficient learning of complex dynamics from governing physics, typically described by partial differential equations (PDEs), using physics-informed neural networks (PINNs). In our experiments, existing versions of PINNs are seen to learn poorly in many problems, especially for complex geometries, as it becomes increasingly difficult to establish appropriate sampling strategy at the near boundary region. Overly dense sampling can adversely impede training convergence if the local gradient behaviors are too complex to be adequately modelled by PINNs. On the other hand, if the samples are too sparse, existing PINNs tend to overfit the near boundary region, leading to incorrect solution. To prevent such issues, we propose a new Boundary Connectivity (BCXN) loss function which provides linear local structure approximation (LSA) to the gradient behaviors at the boundary for PINN. Our BCXN-loss implicitly imposes local structure during training, thus facilitating fast physics-informed learning across entire problem domains with order of magnitude sparser training samples. This LSA-PINN method shows a few orders of magnitude smaller errors than existing methods in terms of the standard L2-norm metric, while using dramatically fewer training samples and iterations. Our proposed LSA-PINN does not pose any requirement on the differentiable property of the networks, and we demonstrate its benefits and ease of implementation on both multi-layer perceptron and convolutional neural network versions as commonly used in current PINN literature.

Reaction-diffusion (Turing) systems are fundamental to the formation of spatial patterns in nature and engineering. These systems are governed by a set of non-linear partial differential equations containing parameters that determine the rate of constituent diffusion and reaction. Critically, these parameters, such as diffusion coefficient, heavily influence the mode and type of the final pattern, and quantitative characterization and knowledge of these parameters can aid in bio-mimetic design or understanding of real-world systems. However, the use of numerical methods to infer these parameters can be difficult and computationally expensive. Typically, adjoint solvers may be used, but they are frequently unstable for very non-linear systems. Alternatively, massive amounts of iterative forward simulations are used to find the best match, but this is extremely effortful. Recently, physics-informed neural networks have been proposed as a means for data-driven discovery of partial differential equations, and have seen success in various applications. Thus, we investigate the use of physics-informed neural networks as a tool to infer key parameters in reaction-diffusion systems in the steady-state for scientific discovery or design. Our proof-of-concept results show that the method is able to infer parameters for different pattern modes and types with errors of less than 10\%. In addition, the stochastic nature of this method can be exploited to provide multiple parameter alternatives to the desired pattern, highlighting the versatility of this method for bio-mimetic design. This work thus demonstrates the utility of physics-informed neural networks for inverse parameter inference of reaction-diffusion systems to enhance scientific discovery and design.

Physics-Informed Neural Networks (PINNs) have been shown to be an effective way of incorporating physics-based domain knowledge into neural network models for many important real-world systems. They have been particularly effective as a means of inferring system information based on data, even in cases where data is scarce. Most of the current work however assumes the availability of high-quality data. In this work, we further conduct a preliminary investigation of the robustness of physics-informed neural networks to the magnitude of noise in the data. Interestingly, our experiments reveal that the inclusion of physics in the neural network is sufficient to negate the impact of noise in data originating from hypothetical low quality sensors with high signal-to-noise ratios of up to 1. The resultant predictions for this test case are seen to still match the predictive value obtained for equivalent data obtained from high-quality sensors with potentially 10x less noise. This further implies the utility of physics-informed neural network modeling for making sense of data from sensor networks in the future, especially with the advent of Industry 4.0 and the increasing trend towards ubiquitous deployment of low-cost sensors which are typically noisier.

In this study, novel physics-informed neural network (PINN) methods for coupling neighboring support points and their derivative terms which are obtained by automatic differentiation (AD), are proposed to allow efficient training with improved accuracy. The computation of differential operators required for PINNs loss evaluation at collocation points are conventionally obtained via AD. Although AD has the advantage of being able to compute the exact gradients at any point, such PINNs can only achieve high accuracies with large numbers of collocation points, otherwise they are prone to optimizing towards unphysical solution. To make PINN training fast, the dual ideas of using numerical differentiation (ND)-inspired method and coupling it with AD are employed to define the loss function. The ND-based formulation for training loss can strongly link neighboring collocation points to enable efficient training in sparse sample regimes, but its accuracy is restricted by the interpolation scheme. The proposed coupled-automatic-numerical differentiation framework, labeled as can-PINN, unifies the advantages of AD and ND, providing more robust and efficient training than AD-based PINNs, while further improving accuracy by up to 1-2 orders of magnitude relative to ND-based PINNs. For a proof-of-concept demonstration of this can-scheme to fluid dynamic problems, two numerical-inspired instantiations of can-PINN schemes for the convection and pressure gradient terms were derived to solve the incompressible Navier-Stokes (N-S) equations. The superior performance of can-PINNs is demonstrated on several challenging problems, including the flow mixing phenomena, lid driven flow in a cavity, and channel flow over a backward facing step. The results reveal that for challenging problems like these, can-PINNs can consistently achieve very good accuracy whereas conventional AD-based PINNs fail.