Chinchun Ooi

Nicholas Sung Wei Yong, Jian Cheng Wong, Pao-Hsiung Chiu et al.

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.

Jian Cheng Wong, Pao-Hsiung Chiu, Chinchun Ooi et al.

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.

Pao-Hsiung Chiu, Jian Cheng Wong, Chinchun Ooi et al.

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.

Jian Cheng Wong, Chinchun Ooi, Abhishek Gupta et al.

A physics-informed neural network (PINN) uses physics-augmented loss functions, e.g., incorporating the residual term from governing partial differential equations (PDEs), to ensure its output is consistent with fundamental physics laws. However, it turns out to be difficult to train an accurate PINN model for many problems in practice. In this paper, we present a novel perspective of the merits of learning in sinusoidal spaces with PINNs. By analyzing behavior at model initialization, we first show that a PINN of increasing expressiveness induces an initial bias around flat output functions. Notably, this initial solution can be very close to satisfying many physics PDEs, i.e., falling into a local minimum of the PINN loss that only minimizes PDE residuals, while still being far from the true solution that jointly minimizes PDE residuals and the initial and/or boundary conditions. It is difficult for gradient descent optimization to escape from such a local minimum trap, often causing the training to stall. We then prove that the sinusoidal mapping of inputs, in an architecture we label as sf-PINN, is effective to increase input gradient variability, thus avoiding being trapped in such deceptive local minimum. The level of variability can be effectively modulated to match high-frequency patterns in the problem at hand. A key facet of this paper is the comprehensive empirical study that demonstrates the efficacy of learning in sinusoidal spaces with PINNs for a wide range of forward and inverse modelling problems spanning multiple physics domains.

Jian Cheng Wong, Chinchun Ooi, Pao-Hsiung Chiu et al.

Physics-Informed Neural Networks (PINNs) have recently shown great promise as a way of incorporating physics-based domain knowledge, including fundamental governing equations, into neural network models for many complex engineering systems. They have been particularly effective in the area of inverse problems, where boundary conditions may be ill-defined, and data-absent scenarios, where typical supervised learning approaches will fail. Here, we further explore the use of this modeling methodology to surrogate modeling of a fluid dynamical system, and demonstrate additional undiscussed and interesting advantages of such a modeling methodology over conventional data-driven approaches: 1) improving the model's predictive performance even with incomplete description of the underlying physics; 2) improving the robustness of the model to noise in the dataset; 3) reduced effort to convergence during optimization for a new, previously unseen scenario by transfer optimization of a pre-existing model. Hence, we noticed the inclusion of a physics-based regularization term can substantially improve the equivalent data-driven surrogate model in many substantive ways, including an order of magnitude improvement in test error when the dataset is very noisy, and a 2-3x improvement when only partial physics is included. In addition, we propose a novel transfer optimization scheme for use in such surrogate modeling scenarios and demonstrate an approximately 3x improvement in speed to convergence and an order of magnitude improvement in predictive performance over conventional Xavier initialization for training of new scenarios.